Current Attempt in Progress
Find the equation of the tangent line to the following curve
at the indicated point.

The area of the figure WXYZ which can be calculated as the sum of the area of the composite triangles ΔXYZ and ΔXWZ is 12 square units

Composite figures are figures comprising of two or more regular figures.

The slope of the side WZ = 2/5

The slope of YZ = -2/1 = -2

The slope of WZ is not the negative inverse of the slope of YZ, therefore, the figure WZ is not perpendicular to YZ and the figure is not a rectangle.

Considering the two triangles formed by the diagonal XZ, we get;

The figure XYZW is a quadrilateral, which is a composite figure comprising of two triangles, triangle ΔXYZ and ΔXWZ

Area of triangle ΔXYZ = (1/2) × 6 × 2 = 6 square units

Area of triangle ΔXWZ = (1/2) × 6 × 2 = 6 square units

The area of the figure = Area of triangle ΔXYZ + Area of triangle ΔXWZ

Area of triangle ΔXYZ + Area of triangle ΔXWZ = 6 + 6 = 12 square units

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The combinations of assembling these rectangles for which it is possible to create a rectangle with the length of 15 and the width 11 with no gaps or overlapping are:

A rectangle is a plane figure with four straight sides and four right angles, especially one with unequal adjacent sides.

Required

Which group forms a rectangle of

[tex]\text{Length}=15[/tex]

[tex]\text{Width}=11[/tex]

First, calculate the area of the big rectangle

[tex]\text{Area}=\text{Length}\times\text{Width}[/tex]

[tex]\text{A}_{\text{Big}}=15\times11[/tex]

[tex]\text{A}_{\text{Big}}=165[/tex]

Next, calculate the area of each rectangle A to E.

[tex]\text{A}_{\text{A}}=11\times7[/tex]

[tex]\text{A}_{\text{A}}=77[/tex]

[tex]\text{A}_{\text{B}}=2\times11[/tex]

[tex]\text{A}_{\text{B}}=22[/tex]

[tex]\text{A}_{\text{C}}=6\times6[/tex]

[tex]\text{A}_{\text{C}}=36[/tex]

[tex]\text{A}_{\text{D}}=6\times5[/tex]

[tex]\text{A}_{\text{D}}=30[/tex]

[tex]\text{A}_{\text{E}}=13\times4[/tex]

[tex]\text{A}_{\text{E}}=52[/tex]

Then consider each option.

(a) 2C + 2D + 2B

[tex]2\text{C}+2\text{D}+2\text{B}=(2\times36)+(2\times30)+(2\times22)[/tex]

[tex]2\text{C}+2\text{D}+2\text{B}=72+60+44[/tex]

[tex]2\text{C}+2\text{D}+2\text{B}=176[/tex]

(b) A + B + C + D

[tex]\text{A}+\text{B}+\text{C}+\text{D}=77+22+36+30[/tex]

[tex]\text{A}+\text{B}+\text{C}+\text{D}=165[/tex]

(c) A + 4B

[tex]\text{A} + 4\text{B}=77+(4\times22)[/tex]

[tex]\text{A} + 4\text{B}=77+88[/tex]

[tex]\text{A} + 4\text{B}=165[/tex]

(d) 3E + 2B

[tex]3\text{E}+2\text{B}=(3\times52)+(2\times22)[/tex]

[tex]3\text{E}+2\text{B}=156+44[/tex]

[tex]3\text{E}+2\text{B}=200[/tex]

(e) E + C + D + 3B

[tex]\text{E} + \text{C} + \text{D} + 3\text{B}=52+36+30+(3\times22)[/tex]

[tex]\text{E} + \text{C} + \text{D} + 3\text{B}=52+36+30+66[/tex]

[tex]\text{E} + \text{C} + \text{D} + 3\text{B}=184[/tex]

Recall that:

[tex]\text{A}_{\text{Big}}=165[/tex]

Only options (b) and (c) match this value.

[tex]\text{A}+\text{B}+\text{C}+\text{D}=165[/tex]

[tex]\text{A} + 4\text{B}=165[/tex]

Hence, options (b) and (c) are correct.

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f(0) = -3

I believe, since the graph has a closed circle/point at (0,-3), f(0) should equal -3. Also, the graph 3x-3 has a domain of x>=0.

However, in terms of limits, the limit approaching x-->0 does not exist since the left and right limits do not equal one another.

Hope this helps.

The graph of the solution to the inequality is attached as image to this answer.

The piece-wise defined function h(x) represents different values of y (the output) depending on the value of x (the input). Each interval of x has a different value assigned to it.

In this particular case, the inequality statements define the intervals for x and their corresponding output values.

Let's break it down:

- For values of x that are greater than -3 and less than or equal to -2, h(x) is assigned the value of -1.

- For values of x that are greater than -2 and less than or equal to -1, h(x) is assigned the value of 0.

- For values of x that are greater than -1 and less than or equal to 0, h(x) is assigned the value of 1.

- For values of x that are greater than 0 and less than or equal to 1, h(x) is assigned the value of 2.

Any values of x outside of these intervals are not defined in this piece-wise function and are typically represented as "not a number" (NaN).

For example, if you were to evaluate h(-2.5), it falls within the first interval (-3 < x ≤ -2), so h(-2.5) would be equal to -1. Similarly, if you were to evaluate h(0.5), it falls within the fourth interval (0 < x ≤ 1), so h(0.5) would be equal to 2.

The graph of the piece-wise function h(x) consists of horizontal line segments connecting the specified values of y for each interval, resulting in a step-like pattern.

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The coefficient of x is -1, indicating that there is one fewer x term compared to the original expression. The constant term is -4, which is the result of subtracting 5 from -9.

To find the difference when subtracting 8x - 5 from 7x - 9, we can use the distributive property to distribute the negative sign to each term in 8x - 5:

(7x - 9) - (8x - 5) = 7x - 9 - 8x + 5

Next, we can combine like terms by adding or subtracting the coefficients of the same variables:

7x - 9 - 8x + 5 = (7x - 8x) + (-9 + 5) = -x - 4

Therefore, the expression that represents the difference when 8x - 5 is subtracted from 7x - 9 is -x - 4.

In this expression, the coefficient of x is -1, indicating that there is one fewer x term compared to the original expression. The constant term is -4, which is the result of subtracting 5 from -9.

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Explanation:

Plug x = -2 into h(x)

h(x) = x+4

h(-2) = -2+4

h(-2) = 2

This means g[ h(-2) ] = g(2) after replacing h(-2) with 2.

g(x) = 5x

g(2) = 5*2

g(2) = 10

Therefore, g[ h(-2) ] = 10

(a) The magnitude of the rod's angular acceleration is (3g/2L) rad/s^2.

(b) The tangential acceleration of the rod's center of mass is (3g/4) m/s^2.

(c) The tangential acceleration of the rod's free end is (3g/2) m/s^2.

(a) To find the magnitude of the rod's angular acceleration, we can use the formula for rotational motion. The torque acting on the rod is due to the gravitational force acting at its center of mass.

The torque is given by τ = Iα, where τ is the torque, I is the moment of inertia, and α is the angular acceleration.

For a thin rod rotating about one end, the moment of inertia is (1/3)ML^2, where M is the mass of the rod and L is its length.

The torque is equal to the product of the gravitational force and the perpendicular distance from the pivot to the center of mass, which is (1/2)L.

So we have τ = (1/2)MgL, where g is the acceleration due to gravity. Substituting these values into the torque equation, we get (1/2)MgL = (1/3)ML^2 α.

Simplifying the equation, we find α = (3g/2L).

Therefore, the magnitude of the rod's angular acceleration is (3g/2L) rad/s^2.

(b) The tangential acceleration of the rod's center of mass can be found using the formula a = αr, where a is the tangential acceleration, α is the angular acceleration, and r is the distance from the center of mass to the pivot point.

In this case, the distance r is (1/2)L, so substituting the values, we get a = (3g/2L)(1/2)L = (3g/4) m/s^2.

Therefore, the tangential acceleration of the rod's center of mass is (3g/4) m/s^2.

(c) The tangential acceleration of the rod's free end is equal to the sum of the tangential acceleration of the center of mass and the product of the angular acceleration and the distance from the center of mass to the free end.

Since the distance from the center of mass to the free end is (1/2)L, the tangential acceleration of the free end is

a + α(1/2)L = (3g/4) + (3g/2L)(1/2)L = (3g/4) + (3g/4) = (3g/2) m/s^2.

Therefore, the tangential acceleration of the rod's free end is (3g/2) m/s^2.

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Answer:

Samuel = 21 orders

Nicole = 31 orders

Miguel = 63 orders

Step-by-step explanation:

Let N represent Nicole's orders, M represents Miguel's orders, and S represent Samuel's orders.

We know that the sum of their tree orders equals 115 as

N + M + S = 115

Since Miguel served 3 times as many orders as Samuel, we know that

M = 3S.

Since Nicole served 10 more orders than Samuel, we know that

N = S + 10

Samuel's Orders:

Now we can plug in 3S for M and S + 10 for N to find S, the number of Samuel's orders:

S + 10 + 3S + S = 115

5S + 10 = 115

5S = 105

S = 21

Thus, Samuel served 21 orders.

Nicole's Orders:

Now we can plug in 21 for S in N = S + 10 to determine how many orders Nicole served:

N = 21 + 10

N = 31

Thus, Nicole served 31 orders.

Miguel's Orders:

Now we plug in 19 for S in M  = 3S to determine how many orders Miguel served:

M = 3(21)

M = 63

Thus, Miguel served 63 orders.

The end of 6 months, Ram should pay Rs. 276250 compounded half-yearly.

Ram borrowed Rs. 250000 from Sit at an interest rate of 21% per annum. To calculate the compound interest, we need to know the compounding period. In this case, the interest is compounded half-yearly, which means it is calculated twice a year.

To find out how much Ram should pay at the end of 6 months, we can use the compound interest formula:

A = P(1 + r/n)^(nt)

Where:
A = the amount to be paid at the end of the time period
P = the principal amount (the initial amount borrowed) = Rs. 250000
r = the interest rate per period (in decimal form) = 21% = 0.21
n = the number of compounding periods per year = 2 (since it's compounded half-yearly)
t = the number of years = 6 months = 6/12 = 0.5 years

Plugging in these values into the formula, we get:

A = 250000(1 + 0.21/2)^(2*0.5)

Simplifying the equation:

A = 250000(1 + 0.105)^(1)

A = 250000(1.105)

A = Rs. 276250

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Answer: 5/ 16

explanation: total= 4x4=16

red and yellow : (r,y) or (y,r)

n= 5

p= 5 1/1 16

p = 5 over 16

Answer:

(8,5)(0,-3)

Step-by-step explanation:

-y=-x+3

y=x-3

Substitute for y:

(x-3)^2-2x=9

x^2-6x+9-2x=9

x^2-8x=0

x(x-8)=0

x=0,8

if x=0,

0-y=3

y=-3

if x=8

8-y=3

-y=-5

y=5

Answer :

x - y = 3

x = 3 + y

y^2 - 2x = 9

y^2 - 2(3+y) = 9

y^2 - 2y -6 -9 = 0

y^2 - 2y -15 = 0

Factorize

y = -3 y = 5

when y = -3

x -(-3) = 3

x = 0

when y = 5

x - 5 = 3

x = 8

Answer:

Yes

Step-by-step explanation:

To know if a table is a function or not, we have to see if 1 input only has 1 output.

Looking at the table each input only has 1 output, so it is a function.

The nth square number of the value given is the squared value of 19, which is 361

The nth square number , S is related by the formula :

Given that , n = 19

The nth square number , where n = 19 can be calculated thus:

S = -19² = 361

S = (-19) * (-19)

S = 361

Therefore, the 19th square number is 361

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The result of dividing (4x³ + 6x² + 20x + 9) by (2x + 1) using long polynomial division is 2x² + 2x + 9 with a remainder of 0.

To divide the polynomial (4x³ + 6x² + 20x + 9) by (2x + 1) using long polynomial division.

Arrange the terms of the dividend and the divisor in descending order of the degree of x:

      2x + 1 | 4x³ + 6x² + 20x + 9

Divide the first term of the dividend by the first term of the divisor and write the result on the top line:

             2x + 1 | 4x³ + 6x² + 20x + 9

                   | 2x²

Multiply the divisor (2x + 1) by the quotient obtained in the previous step (2x²) and write the result below the dividend:

             2x + 1 | 4x³ + 6x² + 20x + 9

           - (4x³ + 2x²)

           ---------------

                        4x² + 20x + 9

Subtract the result obtained in the previous step from the dividend and bring down the next term.

             2x + 1 | 4x³ + 6x² + 20x + 9

           - (4x³ + 2x²)

           ---------------

                        4x² + 20x + 9

                      - (4x² + 2x)

                      ---------------

                               18x + 9

Repeat the process by dividing the term brought down (18x) by the first term of the divisor (2x):

             2x + 1 | 4x³ + 6x² + 20x + 9

           - (4x³ + 2x²)

           ---------------

                        4x² + 20x + 9

                      - (4x² + 2x)

                      ---------------

                               18x + 9

                             - (18x + 9)

                             ---------------

                                         0

The division is complete when the degree of the term brought down becomes less than the degree of the divisor.

In this case, the degree of the term brought down is 0 (a constant term). Since we can no longer divide further, the remainder is 0.

Therefore, the result of the division is:

Quotient: 2x² + 2x + 9

Remainder: 0

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The value of b in the Problem given is 0.5

The given problem can be represented mathematically as below :

We can find be in the expression thus :

4 = 8b

divide both sides by 8 in other to isolate b

4/8 = 8b/8

0.5 = b

Therefore, value of b in the expression is 1/2.

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The length of u, to the nearest inch, is 1818 inches.

To solve this problem, we can use the Law of Sines, which states that the ratio of the length of a side of a triangle to the sine of its opposite angle is constant.

In this case, we'll use the following formula:

a/sin(A) = b/sin(B) = c/sin(C)

Let's label the sides and angles of the triangle:

Side a = u (length of u)

Side b = t (820 inches)

Side c = v (length of v)

Angle A = m/U (132°)

Angle B = m2V (25°)

Angle C = 180° - A - B (as the sum of angles in a triangle is 180°)

Now, we can use the Law of Sines to set up the equation:

u/sin(A) = t/sin(B)

Plugging in the given values:

u/sin(132°) = 820/sin(25°)

To find the length of u, we'll solve this equation for u.

u = (820 [tex]\times[/tex] sin(132°)) / sin(25°)

Using a calculator, we can evaluate the right side of the equation to get the approximate value of u:

u ≈ (820 [tex]\times[/tex] 0.9397) / 0.4226

u ≈ 1817.54 inches

Rounding to the nearest inch, we have:

u ≈ 1818 inches

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,Alice, Raul, and Maria can make 1 whole batch of a dozen cookies together using the ingredients they brought.

To determine how many whole batches of a dozen cookies they can make, we need to compare the amount of flour and butter they have with the required amounts for each batch of cookies.

Let's calculate the total amount of flour and butter each student brought:

Alice:

Flour: 2 cups

Butter: 1/4 cup

Maria:

Flour: 1 1/4 cups

Butter: 3/4 cup

Now let's compare the amounts with the required ingredients for a batch of cookies:

Required amount per batch:

Flour: 3/4 cup

Butter: 1/3 cup

First, let's see how many batches of cookies Alice can make:

Flour: Alice has 2 cups, and each batch requires 3/4 cup. So she can make 2 cups / (3/4 cup) = 8/3 = 2 and 2/3 batches of cookies.

Butter: Alice has 1/4 cup, and each batch requires 1/3 cup. Since she doesn't have enough butter for a full batch, she can't make any batches of cookies with the butter she brought.

Next, let's see how many batches of cookies Maria can make:

Flour: Maria has 1 1/4 cups, and each batch requires 3/4 cup. So she canmake (5/4 cups) / (3/4 cup) = 5/3 = 1 and 2/3 batches of cookies.

Butter: Maria has 3/4 cup, and each batch requires 1/3 cup. So she can make (3/4 cup) / (1/3 cup) = 9/4 = 2 and 1/4 batches of cookies.

Now, to find the maximum number of whole batches they can make together, we take the minimum of the number of batches each student can make:

Alice: 2 and 2/3 batches

Maria: 1 and 2/3 batches

Raul: Not mentioned, so we assume he brought the required ingredients.

The minimum value is 1 and 2/3 batches, which means they can make 1 whole batch of a dozen cookies together.

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The postulate or theorem that proves that two triangles are congruent if two angles and the included side of one triangle are congruent to two angles and the included side of another triangle. Of the above choices, only ASA satisfies this condition. So the answer is (C).

ASA Congruence Theorem explains that if two angles and the included side of one triangle are congruent to two angles and the included side of another triangle, then the two triangles are congruent.

This theorem is a part of triangle congruence criteria in Euclidean geometry. It states that if two angles and the included side of one triangle are equal to the corresponding two angles and included side of another triangle, then the two triangles are congruent, meaning they have the same shape and size.

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Answer:

x = 10

Step-by-step explanation:

the figure inscribed in the circle is a cyclic quadrilateral , all 4 vertices lie on the circumference.

the opposite angles in a cyclic quadrilateral sum to 180° , that is

6x + 1 + 10x + 19 = 180

16x + 20 = 180 ( subtract 20 from both sides )

16x = 160 ( divide both sides by 16 )

x = 10

Answer:

(-3, -2) ∪ (1, ∞)

Step-by-step explanation:

Given inequality:

[tex]\dfrac{x^2+5x+6}{x-1} > 0[/tex]

Begin by factoring the denominator:

[tex]\begin{aligned}x^2+5x+6&=x^2+2x+3x+6\\&=x(x+2)+3(x+2)\\&=(x+3)(x+2)\end{aligned}[/tex]

Therefore, the factored inequality is:

[tex]\dfrac{(x+3)(x+2)}{x-1} > 0[/tex]

Determine the critical points - these are the points where the rational expression will be zero or undefined.

The rational expression will be zero when the numerator is zero:

[tex](x+3)(x+2)=0 \implies x=-3,\;x=-2[/tex]

Therefore, -3 and -2 are critical points.

The rational expression will be undefined when the denominator is zero:

[tex]x-1=0 \implies x=1[/tex]

Therefore, 1 is a critical point.

So the critical points are -3, -2 and 1.

Create a sign chart, using open dots at each critical point (the inequality is greater than, so the interval doesn't include the values).

Choose a test value for each region, including one to the left of all the critical values and one to the right of all the critical values.

Chosen test values: -4, -2.5, 0, 2

For each test value, determine if the function is positive or negative:

[tex]x=-4 \implies \dfrac{(-4+3)(-4+2)}{-4-1} = \dfrac{(-)(-)}{(-)}=\dfrac{(+)}{(-)}=-[/tex]

[tex]x=-2.5 \implies \dfrac{(-2.5+3)(-2.5+2)}{-2.5-1} = \dfrac{(+)(-)}{(-)}=\dfrac{(-)}{(-)}=+[/tex]

[tex]x=0 \implies \dfrac{(0+3)(0+2)}{0-1} = \dfrac{(+)(+)}{(-)}=\dfrac{(+)}{(-)}=-[/tex]

[tex]x=2 \implies \dfrac{(2+3)(2+2)}{2-1} = \dfrac{(+)(+)}{(+)}=\dfrac{(+)}{(+)}=+[/tex]

Record the results on the sign chart for each region (see attached).

As we need to find the values for which the rational expression is greater than zero, shade the positive regions on the sign chart (see attached). These regions are the solution set.

Remember that the intervals of the solution set should not include the critical points, as the critical points of the numerator make the expression zero, and the critical point of the denominator makes the expression undefined. The intervals of the solution set are those where the rational expression is greater than zero only.

Therefore, the solution set is:

-3 < x < -2  or  x > 1

As interval notation:

(-3, -2) ∪ (1, ∞)

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Answer:

Yes

Step-by-step explanation:

Yes, it is a function. If you perform the vertical-line test, the line only touches a point once.

Answer: 21.15 ft²

Step-by-step explanation:

We can use the formula for the area of a triangle:

(b×h)/2

In this case, the base is 9 and the height is 4.7.

So, we substitute the variables with the numbers in this problem.

(9 × 4.7)/2

9 × 4.7 = 42.3

42.3/2 = 21.15

So, our final answer is 21.15 ft²

Savings accounts and CDs are good options for people who want to save money without taking on a lot of risk.

If Anita and Miguel do not take any money from their accounts, Anita's account will grow faster than Miguel's.

This is because the interest rate for Anita's account is 6%, while Miguel's is 5%.

The interest rate is the percentage of the principal that a bank or other financial institution pays for the use of money.

It can be thought of as a fee charged for borrowing money.

The higher the interest rate, the more money a person can earn on their investment.

Anita and Miguel's accounts are probably savings accounts or CDs, which are low-risk investments that pay a fixed interest rate.

Savings accounts and CDs are good options for people who want to save money without taking on a lot of risk.

Anita and Miguel's accounts are probably savings accounts or CDs, which are low-risk investments that pay a fixed interest rate.

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1) The probability that 5 will be working is: 0.187

2) The probability that at least one machine would be working is: 0.006

3) The probability that all would be working is : 1

We are given the parameters as:

Total number of machines = 200

Probability that a Machine is working = 12% = 0.12

1) Now, you want to pick 40 machines and want to find the probability that 5 will be working.

This probability is given by the expression:

P(5 working) = C(40,5) * 0.12⁵·0.88³⁵ ≈ 0.187

where C(n, k) = n!/(k!(n-k)!)

2) The probability that at least one machine would be working is:

0.88⁴⁰ ≈ 0.006

3) The probability that all would be working is the complement of the probability that all have failed. Thus:

P(all working) = 1 - 0.12⁴⁰ ≈ 1

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When x = 3, the expression 2x - 50 evaluates to -44.

To demonstrate an example using the expression 2(x + 5) - 5 × 12, let's simplify it step by step:

Start with the given expression.

2(x + 5) - 5 × 12

Apply the distributive property.

2x + 2(5) - 5 × 12

Simplify within parentheses and perform multiplication.

2x + 10 - 60

Combine like terms.

2x - 50

The simplified form of the expression 2(x + 5) - 5 × 12 is 2x - 50.

Let's consider an example for substituting a value for the variable x:

Suppose we want to evaluate the expression when x = 3. We substitute x = 3 into the simplified expression:

2(3) - 50

Now, perform the calculations:

6 - 50

The result is -44.

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Question

evaluate the expression 2(x+5)-5 x 12.

The statement about the zeros of function g is that they are located at x = 5 + √2 and x = 5 - √2.

When the vertex of function f is moved 4 units to the right and 2 units down, the equation of function g can be represented as g(x) = -(x-5)^2 + 2.

To determine the statement about the zeros of function g, we need to find the x-values where g(x) equals zero.

Setting g(x) = 0 and solving for x:

[tex]0 = -(x-5)^2 + 2[/tex]

Adding (x-5)^2 to both sides:

[tex](x-5)^2 = 2[/tex]

Taking the square root of both sides (considering both positive and negative roots):

x - 5 = ±√2

Adding 5 to both sides:

x = 5 ± √2

Therefore, the zeros of function g are x = 5 + √2 and x = 5 - √2.

In summary, the statement about the zeros of function g is that they are located at x = 5 + √2 and x = 5 - √2.

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Answer: Step-by-step work:

40,328

77

3,105 (Carry the 3)

27,468

302,976

1,609,432

Add the numbers horizontally:

2,943,941

So, 40,328 * 77 = 2,943,941

The remainder when divided by 10 is:

2,943,941 % 10 = 1

Therefore, the remainder is 1.

More concisely:

40,328 * 77 = 2,943,941

2,943,941 % 10 = 1

So the remainder when 2,943,941 is divided by 10 is 1.

Hope this helps! Let me know if you have any other questions

Step-by-step explanation:

The approximate mean of the number of cars sold by the salesman in the past 10 weeks is 4.1.

To find the approximate mean, first, we add all the numbers in the data set, and then we divide that sum by the total number of values in the data set.

The formula for finding the approximate mean is as follows: Approximate mean = sum of the values in the data set / total number of values in the data set.

The following data set is given: Number of cars sold by a salesman in the past 10 weeks: 3, 5, 2, 4, 7, 5, 6, 3, 2, 4.

To find the approximate mean, we first need to add all the values: 3 + 5 + 2 + 4 + 7 + 5 + 6 + 3 + 2 + 4 = 41 The sum of all the values is 41.

Next, we need to divide this sum by the total number of values in the data set. In this case, the total number of values is 10. Therefore, the approximate mean for the given data set is: Approximate mean = 41 / 10 = 4.1

Therefore, the approximate mean of the number of cars sold by the salesman in the past 10 weeks is 4.1.

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Answer:

a=55°

b=123°

c=55°

d=123°

Type the expressions as radicals y^5/2.

[tex] \sqrt{ {y}^{5} } [/tex]

Radical:- The ( √ ) symbol that is used to denote square root or nth roots...

It can be rewritten as rational exponents ( exponents which are fractions ) using the formula:-

[tex] \sqrt[n]{x} = {x}^{ \frac{1}{n} } [/tex]

Generally, using the power rule of exponents:

[tex] \sqrt[n]{ {x}^{m} } = {( {x}^{m)} }^{ \frac{1}{n} } = {x}^{ \frac{m}{n} } [/tex]

Let's take an example to understand better:

• convertion between radicals and rational exponents:

[tex] \sqrt[7]{ {8}^{4} } = {8}^{ \frac{4}{7} } [/tex]

Since the type of radical corresponds with the denominator of a rational exponent, we know the denominator of the exponent will be 7 ..

[tex] {y}^{ \frac{5}{2} } = \sqrt{ {y}^{5} } [/tex]

As , √ denotes ½ ..

The expression "y 5/2" can be written as the fifth root of y squared: √[[tex]y^{2}[/tex]]^(1/5).

The expression "y 5/2" can be written as the fifth root of y squared: √([tex]y^{2}[/tex])^(1/5).

To explain this, let's break it down:

The numerator, [tex]y^{2}[/tex], represents y raised to the power of 2.

Taking the square root of [tex]y^{2}[/tex] simplifies it to √([tex]y^{2}[/tex]).

Finally, raising the result to the power of 1/5 gives us the fifth root of y squared: √([tex]y^{2}[/tex])^(1/5).

In other words, the expression "y 5/2" represents the operation of first squaring y, then taking the fifth root of the resulting value. This is equivalent to finding the value that, when raised to the power of 5, yields [tex]y^{2}[/tex].

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