Solving exponential and logarithmic equations requires a strong understanding of the properties of exponents and logarithms, as well as the use of logarithmic identities and inverse operations. It is important to simplify expressions and isolate the variable using basic algebraic operations, logarithmic identities, and inverse functions.
When solving exponential or logarithmic equations, there are several important steps to consider:
Simplify the equation using basic algebraic operations, such as multiplication, division, addition, and subtraction.
Isolate the exponential or logarithmic expression on one side of the equation.
Use the properties of exponents or logarithms to simplify the expression, if possible.
Use logarithmic identities such as logarithmic rules and change-of-base formula to simplify the expression.
Solve for the variable by using inverse operations.
Check your answer by substituting it back into the original equation and verifying that both sides are equal.
Identities and logs can be helpful in solving exponential and logarithmic equations because they provide a set of rules that can be used to simplify expressions. For example, logarithmic identities such as log(a * b) = log(a) + log(b) and log(a/b) = log(a) - log(b) can be used to simplify expressions containing products and quotients.
However, identities and logarithms can also be confusing because they involve complex concepts and require a solid understanding of logarithmic properties. This can make solving exponential and logarithmic equations challenging for some students.
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when the results of experimentation or historical data are used to assign probability values, the method used to assign probabilities is referred to as the . a. subjective method b. classical method c. relative frequency method d. posterior method
The method used to assign probabilities when the results of experimentation or historical data are used to assign probability values is referred to as the relative frequency method.
The relative frequency method is a way to assign probability values to events based on how often they have occurred in the past.
The probability of an event is determined by counting the number of times the event has occurred in a given set of observations, and dividing that number by the total number of observations. For example, if a certain event has occurred 20 times out of 100 total observations, the probability of that event would be 0.2 (20/100). This method is based on the idea that the probability of an event is related to its frequency of occurrence in a large number of trials.
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Select the correct answer. Which is the correct solution to the expression 3 + 5^2? You can use a calculator to find the answer. A. 10 B. 13 C. 28 D. 64
Answer:28
Step-by-step explanation:
5x5=25
25+3=28
Answer:
C. 28
Step-by-step explanation:
3 + 5^2
3 + 25 = 28
27+____=44
30+____=44
[tex]27 + g = 44 \\ 30 + g = 44[/tex]
How do you solve these equations to set them equal?
The value of g in 27+g=44 is 17 and the value of g in 30+g=44 is 14.
How to solve the mathematical equations:Multiplication, division, addition, and subtraction. A mathematical equation's two sides will remain equal if we add the same integer to each of them.
The definition of an equation in algebra is a mathematical statement that demonstrates the equality of two mathematical expressions. For instance, the equation 3x + 5 = 14 consists of the two equations 3x + 5 and 14, which are separated by the equal sign.
A mathematical statement known as an equation is made up of two expressions joined together by the equal sign. A formula would be 3x - 5 = 16, for instance. When this equation is solved, we discover that the value of the variable x is 7.
Equations come in two varieties: identities and conditional equations. All possible values of the variables result in an identity. Only certain combinations of the variables' values make a conditional equation true.
= 27 + g = 44
= 27 - 44 = g
g = 17
= 30 + g = 44
= 30 - 44 = g
g = 14
Therefore, the value of g in 27+g=44 is 17 and the value of g in 30+g=44 is 14.
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A concrete beam is to rest on two concrete pillars. The beam is a cuboid with sides of length 0.5m, 3m and 0.4m. The pillars have diameter 0.4m and height 2m. Calculate the total volume of concrete needed to make the beam and the pillars. Round your answer to a sensible level of accuracy.
In a case whereby concrete beam is to rest on two concrete pillars. The beam is a cuboid with sides of length 0.5m, 3m and 0.4m with diameter 0.4m and height 2m. the total volume of concrete needed to make the beam and the pillars is 1.103m^3
How can the concrete needed to make the beam be calculated?The concept here is total volume , then we can calculate this for concrete needed to make the beam and the pillars as:
Vtd = V1 + V2
= V1= lwh
where length l = 0.5m,
width w= 3m
height h= 0.4m.
V1 + V2 = (0.5m* 3m * 0.4m) + 2(π/4 d^2. h)
where h =height 2m
Then V1 + V2 = 1.1026m^3
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X - 4.65 less than or equal to 0.09
it's an equation not a question
The required given expression is a equation x - 4.65 = 0.09.
What is Equation?The definition of an equation in algebra is a mathematical statement that demonstrates the equality of two mathematical expressions. For instance, the equation 3x + 5 = 14 consists of the two equations 3x + 5 and 14, which are separated by the 'equal' sign.
According to question:In any expression variables and constants are expressed by equal sign,
Then the expression is called equaion.
So, as question says,
x - 4.65 = 0.09
x = 4.74
Thus, It is a equation of value 4.74.
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the length of the mid segment is:
Check the picture below.
so from the picture below we can say that parallel sides or bases of the trapezoid are AB and DC, that means our midsegment is found halfway of BC and halfway of AD, let's get the midpoints for those and then get their length.
[tex]~~~~~~~~~~~~\textit{middle point of 2 points } \\\\ A(\stackrel{x_1}{0}~,~\stackrel{y_1}{3})\qquad D(\stackrel{x_2}{2}~,~\stackrel{y_2}{0}) \qquad \left(\cfrac{ x_2 + x_1}{2}~~~ ,~~~ \cfrac{ y_2 + y_1}{2} \right) \\\\\\ \left(\cfrac{ 2 +0}{2}~~~ ,~~~ \cfrac{ 0 +3}{2} \right) \implies \left(\cfrac{ 2 }{2}~~~ ,~~~ \cfrac{ 3 }{2} \right)\implies (1~~,~~\frac{3}{2}) \\\\[-0.35em] ~\dotfill[/tex]
[tex]~~~~~~~~~~~~\textit{middle point of 2 points } \\\\ B(\stackrel{x_1}{2}~,~\stackrel{y_1}{5})\qquad C(\stackrel{x_2}{6}~,~\stackrel{y_2}{4}) \qquad \left(\cfrac{ x_2 + x_1}{2}~~~ ,~~~ \cfrac{ y_2 + y_1}{2} \right) \\\\\\ \left(\cfrac{ 6 +2}{2}~~~ ,~~~ \cfrac{ 4 +5}{2} \right) \implies \left(\cfrac{ 8 }{2}~~~ ,~~~ \cfrac{ 9 }{2} \right)\implies (4~~,~~\frac{9}{2})[/tex]
let's get their distance
[tex]~~~~~~~~~~~~\textit{distance between 2 points} \\\\ (\stackrel{x_1}{1}~,~\stackrel{y_1}{\frac{3}{2}})\qquad (\stackrel{x_2}{4}~,~\stackrel{y_2}{\frac{9}{2}})\qquad \qquad d = \sqrt{( x_2- x_1)^2 + ( y_2- y_1)^2} \\\\\\ d=\sqrt{(4-1)^2 + (\frac{9}{2}-\frac{3}{2})^2}\implies d=\sqrt{3^2 + 3^2}\implies {\Large \begin{array}{llll} \stackrel{midsegment}{ d=\sqrt{18}} \end{array}}[/tex]
In ΔIJK, k = 1.9 inches, m∠J=59° and
m∠K=18°. Find the length of j, to the nearest 10th of an inch.
Check the picture below.
[tex]\textit{Law of sines} \\\\ \cfrac{sin(\measuredangle A)}{a}=\cfrac{sin(\measuredangle B)}{b}=\cfrac{sin(\measuredangle C)}{c} \\\\[-0.35em] ~\dotfill\\\\ \cfrac{\sin(18^o)}{1.9}=\cfrac{\sin(59^o)}{j}\implies j=\cfrac{1.9\sin(59^o)}{\sin(18^o)}\implies j\approx 5.3~in[/tex]
Make sure your calculator is in Degree mode.
The length of side J is 5.6 inches.
What is a triangle?A triangle is a 2-D figure with three sides and three angles.
The sum of the angles is 180 degrees.
We can have an obtuse triangle, an acute triangle, or a right triangle.
We have,
To find the length of side J, we can use the law of sines, which states that for any triangle with sides a, b, and c opposite to angles A, B, and C respectively:
a/sin(A) = b/sin(B) = c/sin(C)
In this case, we know that side K is 1.9 inches, and angles J and K are 59° and 18° respectively.
Let's use the law of sines to find the length of side J:
J/sin(59°) = 1.9/sin(18°)
J = (sin(59°)/sin(18°)) * 1.9
J ≈ 5.6 inches
(rounded to the nearest tenth of an inch)
Therefore,
The length of side J is approximately 5.6 inches.
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what is the answer to 12+13
Answer: 25
Step-by-step explanation: 10+10=20 and 2+3=5. 20+5=2
Can someone please help
Find the average if another set of three consecutive numbers. What do you notice? Explain why the thing you notice must always work or find a counterexample
The average of three consecutive numbers 17, 18, 19 is 18 and the average of three consecutive numbers -100, -99, -98 is -99
Consider the three consecutive numbers 17, 18, 19.
We find the average number would be found by adding all the terms and dividing by 3:
(17 + 18 + 19)/3
= (54)/3
= 18
consider another example: three consecutive numbers are -100, -99, -98.
Their average would be,
[-100 + (-99) + (-98) ] ÷ 3
= -297 ÷ 3
= -99
We can observe that the average of three consecutive numbers is the second term given in that set.
Let m, m + 1, m + 2 be set of any three consecutive numbers.
The average of these three consecutive numbers would be,
A = [m + (m + 1) + (m + 2)] ÷ 3
A = (3m + 3) ÷ 3
A = 3(m + 1) ÷ 3
A = m + 1
Thus, the rule will always work, as we know that average is the central value of a set of numbers.
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The complete question is:
Consecutive numbers follow one right after the other. An example of three consecutive numbers is 17, 18, and 19. Another example is -100, -99, -98. Choose any set of three consecutive numbers. Find their average. What do you notice? Find the average of another set of three consecutive numbers. What do you notice? Explain why the thing you noticed must always work, or find a counterexample.
if the diameter of the output cylinder of a imple hydrolic ytem were greater how would thi affect the output force?
If the diameter of the output cylinder of a simple hydraulic system were greater, this would increase the output force.
In a simple hydraulic system, a small force applied to a small piston (input cylinder) is amplified and transferred to a larger piston (output cylinder) through a fluid. The output force is equal to the input force multiplied by the ratio of the areas of the two pistons. The area of a cylinder is equal to the radius squared times pi. So, if the diameter of the output cylinder is increased, the area of the output cylinder will increase and the output force will be greater.
It's important to note that increasing the diameter of the output cylinder will also increase the volume of the output cylinder, so more fluid will be required to fill the cylinder and the system may need a larger fluid pump or reservoir.
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Which of the following statements is NOT true regarding an expression written in scientific notation in the form of a times 10 Superscript n baseline?
A. The value of a must be greater than or equal to 1 and smaller than 10.
B. The value of n must be an integer.
C. Doubling n results in a doubling of the value of the expression.
D. Doubling a results in a doubling of the value of the expression.
How do I do B for 17 and 18 I really need help
17)
The situation models a function, the independent variable is the number of plans and the dependent variable is the cost.
b) The domain of the function is {x:x≤4} while the range is {y:y≤120}
18) The independent variable is the miles travelled and the dependent variable is the cost.
b) The domain of the function is {x:x≤20} and the range is {y:y≤72.8}
What is the domain and the range?We know that the domain of the function represents all of the values can be assigned to the independent variable in the function while the range has to do with all the values that the functions can be able to return.
In this case, we have real life models and we are told to convert them into functions;
a) y = 30 x
b) y = 2.80 + 3.50x
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Does someone mind helping me with this question? Thank you!
Answer:
80 pennies
160 nickels
Step-by-step explanation:
.01p + .05n = 8.80 multiply through by 100
p + 5n = 880
2p = n
substitute 2p for n is the bold equation
p + 5(2p) = 880
p + 10p = 880
11p = 880 Divide both sides by 11
p = 80
2p = n
2(80) = n
160 = n
Check:
(.01)(80) + (.05)(160) = 8.80
0.80 + 8 = 8.80
8.80 = 8.80 Checks
a manufacturer of banana chips would like to know whether its bag filling machine works correctly at the 436 gram setting. is there sufficient evidence at the 0.05 level that the bags are underfilled? assume the population is normally distributed.
Alternative Hypothesis is sufficient evidence at the 0.05 level that the bags are underfilled.
In the given question, we have to explain that there is sufficient evidence at the 0.05 level that the bags are underfilled or not assuming that the population is normally distributed.
We can reject the null hypothesis if the sample contains sufficient data to refute the assertion that there is no effect in the population (p). Or else, we are not able to rule out the null hypothesis.
The claim in the alternative hypothesis that you anticipate or hope to be accurate.
The complement of the null hypothesis is the alternative hypothesis. The extensive nature of null and alternative hypotheses ensures that they account for all potential outcomes.
Since, bag filling machine works correctly at the 436 gram setting. Test the alternative hypothesis in place of the claim that the true mean is less than 436. This test has a left tail.
The null and alternative hypothesis are
H(0): μ = 436
H(a): μ < 436
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. Zachariah has 9 American stamps out of 15 stamps. Khai has 13 American stamps out of 20 stamps. Which statement is correct?
Due to the fact that 8 over 14 is more than 11 over 21, Zachariah has a higher ratio of American stamps than Khai.
What is a fraction?Due to the fact that 8 over 14 is more than 11 over 21, Zachariah has a higher ratio of American stamps than Khai.
An element of a number or any number of equal pieces is represented by a fraction.
Given that Khai and Zachariah are stamp collectors.
Zachariah also contains eight of the fourteen American stamps that are available.
Khai contains 11 American stamps out of a total of 21.
Khai equals 8/14 Zachariah equals 11/21
Zachariah has a higher proportion of American stamps than Khai does since 8 over 14 is more than 11 over 21.
After all, we would convert 11/21 and 8/14 into fractions.
Tell them that we would turn their fractional values into decimals and then into percentiles.
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There are 150,000 households in Market City. A local phone repair shop takes a random sample of 50 households and finds that the average number of phones per household from the sample last year that needed to be repaired was 1.05 ± 0.23. Which of the following is an estimate of the total number of phones that needed repairing last year in Market City?
The estimate of the total number of phones that needed repairing last year in Market City is given as follows:
Between 123,000 and 157,500.
How to obtain the estimate?The estimate is obtained applying the proportions in the context of this problem.
The confidence interval is given as follows:
1.05 ± 0.23.
Hence the bounds of the interval are given as follows:
1.05 - 0.23 = 0.82.1.05 + 0.23 = 1.28.Then the amounts out of 150,000 households are given as follows:
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Write the equation of a line perpendicular to y=2x−5 that passes through the point (2, -5) in slope-intercept form.
To write the equation of a line that is perpendicular to y = 2x - 5, we need to find the slope of the new line.
As we know, the slope of a line perpendicular to y = 2x - 5 is the negative reciprocal of the slope of y = 2x - 5, which is -1/(2) = -1/2.
To find the equation of the line in slope-intercept form (y = mx + b), we can use the point (2, -5) and the slope of -1/2.
We know that the point (2, -5) belongs to the line, so we can substitute these values into the equation:
y = -1/2x + b
-5 = -1/2(2) + b
-5 = -1 + b
b = -4
So, the equation of the line in slope-intercept form is:
y = -1/2x -4
This is the equation of a line that is perpendicular to y = 2x - 5 and passes through the point (2, -5)
Lane Games
ChatGPT Jan 9 Version. Free Research Preview. Our goal is to ma
Solve for r²
1
S= Ar^2h
Answer:
I think r^2= 4S/πh
Step-by-step explanation:
S=1/4πr^2h
Multiple 4 on both sides
4S=πr^2h
Divide π and h on both sides
r^2=4S/πh
Sorry if this wrong
I Need Help I’m Desperate i really need it cause it due tmr
The slope is 1.33, the y-intercept is -1 and the equation is y = 1.333x -1.
The solution has been obtained using the slope intercept form.
What is slope intercept form?
The line with slope m and y-intercepts b is the graph of the linear equation y = mx + b. The values of m and c are real integers, and the form of the linear equation is known as the slope-intercept form.
A line's steepness is indicated by its slope, m. Sometimes, a line's slope is also referred to as its gradient. The y-coordinate of the place where the line's graph contacts the y-axis is represented by a line's y-intercept.
We are given graph with straight line.
The two points on the graph are (0,-1) and (3,3)
Let's find the slope(m)
[tex]m=\frac{y_{2} -y_{1} }{x_{2} -x_{1} }[/tex]
[tex]m=\frac{3+1 }{3 -0 }[/tex]
m = 4/3
m = 1.333
We know that b = y₁ - x₁(y₂ - y₁)/(x₂ - x₁)
So,
⇒b = (-1) - 0(3 + 1)/(3 - 0)
⇒b = -1 + 0
⇒b = -1
The equation comes out to be
⇒y = mx + b
⇒y = 1.333x -1
Hence, the slope is 1.33, the y-intercept is -1 and the equation is
y = 1.333x -1.
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Since, there are multiple questions, so the question answered above is attached below
The average of five numbers is 18. Let the first number be increased by 1, the second number by 2, the third number by 3, the fourth number by fourth, and the fifth number by 5. What is the average of the set of increased numbers?
The average of the increased set of numbers is 21
What is the average of a set of numbers?The average, also known as the arithmetic mean is found by adding together, the numbers in a set of numbers, and dividing the sum by the count of the numbers.
The average of the 5 numbers = 18
The amount by which the numbers are increased are as follows;
First number; Increased by one
Second number; Increased by two
Third number; Increased by three
Fourth number; Increased by four
Fifth number; Increased by five
Let a represent the first number, b the second number, c the third number, d, the fourth number, e, the fifth number, we get;
The increased numbers are; a + 1, b + 2, c + 3, d + 4, e + 5
The initial average can be presented as follows;
Initial average = [tex]\frac{a+b+c+d+e}{5} = 18[/tex]
The imncreased average becomes;
New average = [tex]\frac{(a+1)+(b+2)+(c+3)+(d+4)+(e+5)}{5} = \frac{a+b+c+d+e+15}{5}[/tex]
The new average = (a + b + c + d + e + 15)/5 = (a + b + c + d + e)/15 + 15/5
The term (a + b + c + d + e)/15 in the expression on the right for the new average is the initial average and is equivalent to 18, therefore;
(a + b + c + d + e)/15 = 18
The new average = (a + b + c + d + e)/15 + 15/5
(a + b + c + d + e)/15 + 15/5 = 18 + 15/5 = 18 + 3
The new average = 18 + 3 = 21
The average of the increased set of numbers (the new average) is 21
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Show full solution (all steps). Consider the following triangle to answer the questions.
a) Identify the hypotenuse side.
b) Determine the missing side using the
Pythagorean Theorem.
b) Determine the perimeter.
c) Determine the area.
Answer:
a) The hypotenuse side is BC.
b) Using the Pythagorean Theorem, we can determine the missing side: AB = √(AC² - BC²) = √(6² - 4²) = √(36 - 16) = √20 = 4.47.
c) The perimeter of the triangle is AC + BC + AB = 6 + 4 + 4.47 = 14.47.
d) The area of the triangle is ½ × AC × AB = ½ × 6 × 4.47 = 13.7.
Step-by-step explanation:
The Pythagorean Theorem states that in a right triangle, the sum of the squares of the two shorter sides is equal to the square of the hypotenuse. So, for the triangle given, we can use this theorem to calculate the length of side AB, which is the missing side. Then, we can calculate the perimeter of the triangle by adding up all three sides and the area of the triangle by using the formula ½ × AC × AB.
in the diagram below of trapezoid RSUT, X is the midpoint and V is the midpoint of SU if RS=30 and XV=44, what is the length of TU
The length of TU in a trapezoid RSUT is 34 units.
How to calculate length of TU in a Trapezoid?To find the length of TU, we can use the midpoint formula. The midpoint formula states that for a line segment with endpoints (x1, y1) and (x2, y2), the midpoint is ((x1 + x2)/2, (y1 + y2)/2).
In this case, we know that X is the midpoint of line segment RS and V is the midpoint of line segment SU. So we can set up the following equation:
(RS/2) + (SU/2) = 44
Where (RS/2) = (30/2) = 15 and (SU/2) = TU/2
So we can substitute this into the equation and get:
15 + (TU/2) = 44
To find TU, we can solve this equation by multiplying both sides by 2:
TU = 2(44) - 2(15) = 64 - 30 = 34
So the length of TU is 34 units.
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Please help
Find the area of the given circle. Round to the nearest tenth
The area of a circle with a radius of 3.5cm rounded to the nearest tenth is 38.5 cm².
What is the area of the circle?A circle is simply a closed 2-dimensional curved shape with no corners or edges.
The area of a circle is expressed mathematically as;
Area = πr²
Where r is radius and π is constant pi ( π = 3.14 )
From the diagram;
Radius r = 3.5cmArea A = ?Plug the given value into the above equation and solve for A.
Area = πr²
Area = 3.14 × ( 3.5cm )²
Area = 3.14 × 12.25 cm²
Area = 38.5 cm²
Therefore, the area of the circle is 38.5 cm².
Option B) 38.5 cm² is the correct answer.
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A plumber has a very long piece of pipe that is used to run city water parallel to a major roadway. The pipe is cut into two sections. One section of pipe is 12 ft. shorter than the other. If 3 of the length of the shorter pipe is 120 ft., how long is the longer piece of the pipe?
If a plumber has a very long piece of pipe that is used to run city water parallel to a major roadway. The length of the longer piece of the pipe is: 172 ft.
How to find the length?Let x represent the longer piece
Let x - 12 represent the shorter piece
So,
(3)(x - 12) = 120
Multiply both sides by 4/3
x - 12 = 120 (4/3)
x - 12 = 160
Add 12 to both sides
x = 172 ft (longer piece)
Therefore the length of the longer piece is 172 ft.
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The correct question is:
A plumber has a very long piece of pipe that is used to run city water parallel to a major roadway. The pipe is cut into two sections. One section of the pipe is 12 ft. shorter than the other. If 3/4 of the length of the shorter pipe is 120 ft., how long is the longer piece of the pipe?
Select all the equations that are true when x = 6.
6x = 12
17 – x = 11
x ÷ 2 = 3
19 ÷ 3 = x
5 + x = 11
What is 0.025m in centimeters?
Answer:2.5cm
Step-by-step explanation:1cm =0.01 m
Function Notation Worksheet #2
Use the functions below to evaluate at the given value.
f(x) = 4x - 7
g(x) = 4*
1. f(-2) =
4.j(-1) =
7. k(-1)-5=
10. f(a - 1) =
j(x) = 2x² - 3
2. g(3) =
5.k(-2) =
8. f(0) + 7 =
11. j(3a) =
k(x) = |x - 3|
h(x) = √x
3. h(25) =
6. g(-2) =
9. g(0) =
12. f(2p - 3q) =
the value of functions f(-2)=-15, g(3) =12.
What is a function?
A unique kind of relation called a function is one in which each input has precisely one output. In other words, the function produces exactly one value for each input value. The graphic above shows a relation rather than a function because one is mapped to two different values. The relation above would turn into a function, though, if one were instead mapped to a single value. Additionally, output values can be equal to input values.
The x-values are input into the function machine. The function machine then performs its operations and outputs the y-values. The function within can be any function.
f(x) = 4x - 7
g(x) = 4*x
f(-2) = 4(-2)-7 = -8-7 = 15
g(3) = 4*3 = 12
Hence the value of functions f(-2)=-15, g(3) =12.
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Find the product (14)(4)
Answer:
56
Step-by-step explanation:
10 times 4 is 40
4 times 4 is 16
40+16=56
Marvin is putting together a business outfit. There are 8 pairs of pants and 7 dress shirts to choose from. How many different outfits can Marvin put together?
Answer:
54
Step-by-step explanation:
This is legit just 8 x 7,
4. Analyze and Persevere What is a power that
has the same value as 18? Explain.
The powers that have same value as 1^8 are 1^n, n^0 where n=real number.
What are real numbers?
Real numbers are simply the combination of rational and irrational numbers, in the number system. In general, all the arithmetic operations can be performed on these numbers and they can be represented in the number line, also. At the same time, the imaginary numbers are the un-real numbers, which cannot be expressed in the number line and are commonly used to represent a complex number. Some of the examples of real numbers are 23, -12, 6.99, 5/2, π, and so on.
Now,
As we know anything raised to the power of 1 is always one and 0 raised to the power of anything is also one.
Hence, the powers that have same value as 1^8 are 1^n, n^0 where n=real number.
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