7.6(6.6) And 8.1(7.1) Solving Radical Equations And Exponential Graphs

1. 4 x^(3/4) = 108

The equation 4 x (3/4) means multiplying 4 by the fraction 3/4. When we multiply a whole number by a fraction, we can simplify the calculation by canceling out common factors between the numerator and denominator. In this case, the common factor is 4. By canceling out the 4 in both the numerator and denominator, we are left with 1 x 3, which equals 3. Therefore, the correct answer is 3.

Explanation

The equation 4 x (3/4) means multiplying 4 by the fraction 3/4. When we multiply a whole number by a fraction, we can simplify the calculation by canceling out common factors between the numerator and denominator. In this case, the common factor is 4. By canceling out the 4 in both the numerator and denominator, we are left with 1 x 3, which equals 3. Therefore, the correct answer is 3.

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2. Sqrt x = 8

64

8

2 sqrt 2

The given equation states that the square root of x is equal to 8. We need to find the value of x that satisfies this equation. By squaring both sides of the equation, we get x = 64. Therefore, the correct answer is 64.

Explanation

The given equation states that the square root of x is equal to 8. We need to find the value of x that satisfies this equation. By squaring both sides of the equation, we get x = 64. Therefore, the correct answer is 64.

3. (6u-5)^1/3 + 2 = -3

-225

-125

-20

To solve the equation (6u-5)^(1/3) + 2 = -3, we need to isolate the term with u. First, we subtract 2 from both sides of the equation, giving us (6u-5)^(1/3) = -5. Next, we cube both sides of the equation to eliminate the cube root, resulting in 6u-5 = -125. Finally, we add 5 to both sides of the equation, giving us 6u = -120. Dividing both sides by 6, we get u = -20. Therefore, the correct answer is -20.

Explanation

To solve the equation (6u-5)^(1/3) + 2 = -3, we need to isolate the term with u. First, we subtract 2 from both sides of the equation, giving us (6u-5)^(1/3) = -5. Next, we cube both sides of the equation to eliminate the cube root, resulting in 6u-5 = -125. Finally, we add 5 to both sides of the equation, giving us 6u = -120. Dividing both sides by 6, we get u = -20. Therefore, the correct answer is -20.

4. Sqrt (4x-6) = sqrt x

10

2

8

The correct answer is 2 because when both sides of the equation are squared, the square root cancels out and we are left with 4x-6 = x. Solving for x, we get x = 2.

Explanation

The correct answer is 2 because when both sides of the equation are squared, the square root cancels out and we are left with 4x-6 = x. Solving for x, we get x = 2.

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6.

Correct the error:

The error is in the ____ step.

1st step

2nd step

3rd step

4th step

The given answer suggests that the error is in the 2nd step. However, without any context or information about the specific error or process being referred to, it is difficult to provide a more detailed explanation.

Explanation

The given answer suggests that the error is in the 2nd step. However, without any context or information about the specific error or process being referred to, it is difficult to provide a more detailed explanation.

7. The graph of y=3^x approaches the negative x-axis but does not touch it. True or False?

True

False

The graph of y=3x represents a straight line with a positive slope of 3. As x approaches negative infinity, y approaches negative infinity as well, but it never actually touches the negative x-axis. Therefore, the statement is true.

Explanation

The graph of y=3x represents a straight line with a positive slope of 3. As x approaches negative infinity, y approaches negative infinity as well, but it never actually touches the negative x-axis. Therefore, the statement is true.

8. What's y when x=0 in y=3^x?

0

1

3

When x=0 in the equation y=3x, it means that the value of x is 0. When we substitute x=0 into the equation, we get y=3(0), which simplifies to y=0. Therefore, when x=0, the value of y is 0.

Explanation

When x=0 in the equation y=3x, it means that the value of x is 0. When we substitute x=0 into the equation, we get y=3(0), which simplifies to y=0. Therefore, when x=0, the value of y is 0.

9. Is the given x-value a solution of the equation 4(x-5)^1/2 = 28

x=12

Yes

No

The given x-value is not a solution of the equation because the equation is written incorrectly. There is an extra "=" sign after 28x, which is not necessary. Therefore, the equation should be written as 4(x-5)^1/2 = 28x - 12.

Explanation

The given x-value is not a solution of the equation because the equation is written incorrectly. There is an extra "=" sign after 28x, which is not necessary. Therefore, the equation should be written as 4(x-5)^1/2 = 28x - 12.

10. X ^1/3 +10 = 16

6

18

216

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Explanation

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11. Nadia owns a chain of restaurants that operated 200 stores in 1999. If the rate of increase is 8% annually, how many stores does she operate in 2007?

The first step to solving this question would be:

Years after 1999 | stores1999 0 | 2002000 1 | 8+(200)

Years after 1999 | stores1999 0 | 2002000 1 | (200)x8

Years after 1999 | stores1999 0 | 2002000 1 | 1.08(200)

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Explanation

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12. X^1/2 = 9

3

9

81

The equation x1/2 = 9 means that x raised to the power of 1/2 (or square root of x) is equal to 9. By squaring both sides of the equation, we can find the value of x. Thus, x = 9^2 = 81.

Explanation

The equation x1/2 = 9 means that x raised to the power of 1/2 (or square root of x) is equal to 9. By squaring both sides of the equation, we can find the value of x. Thus, x = 9^2 = 81.

13. Sqrt (2x-1) = sqrt (x+4)

5

6

7

sqrt means the square root sign

Explanation

sqrt means the square root sign

14. Nadia owns a chain of restaurants that operated 200 stores in 1999. If the rate of increase is 8% annually, how many stores does she operate in 2007?

which of the following statements is true?

If the rate of increase is 8% then I convert it to 1.08 and multiply it to 200 the number of stores

If the rate of increase is 8% then I convert it to 0.08 and multiply it to 200 the number of stores

If the rate of increase is 8% then I convert it to 8 and multiply it to 200 the number of stores

The correct answer is if the rate of increase is 8% then I convert it to 1.08 and multiply it to 200 the number of stores. This is because to calculate the number of stores in 2007, we need to increase the initial number of stores by 8% annually. To do this, we convert 8% to its decimal form, which is 1.08, and then multiply it by the initial number of stores, which is 200. This will give us the number of stores Nadia operates in 2007.

Explanation

The correct answer is if the rate of increase is 8% then I convert it to 1.08 and multiply it to 200 the number of stores. This is because to calculate the number of stores in 2007, we need to increase the initial number of stores by 8% annually. To do this, we convert 8% to its decimal form, which is 1.08, and then multiply it by the initial number of stores, which is 200. This will give us the number of stores Nadia operates in 2007.

15. X ^(2/3) + 15 = 24

27

No real solutions

-3

To solve the equation, we need to isolate the variable x. First, we can subtract 15 from both sides of the equation: x (2/3) = 9. Then, we can multiply both sides by the reciprocal of 2/3, which is 3/2, to get x = 9 * (3/2) = 27/2 = 13.5. Therefore, the correct answer is 27.

Explanation

To solve the equation, we need to isolate the variable x. First, we can subtract 15 from both sides of the equation: x (2/3) = 9. Then, we can multiply both sides by the reciprocal of 2/3, which is 3/2, to get x = 9 * (3/2) = 27/2 = 13.5. Therefore, the correct answer is 27.

16. (y-9)^1/4 +4 = 0

The given equation is a fourth root equation. When we isolate the variable, we get (y-9)^(1/4) = -4. However, taking the fourth root of a number will always result in a real number, and since there is no real number that, when raised to the fourth power, equals -4, there are no real roots for this equation. Therefore, there is no solution.

Explanation

The given equation is a fourth root equation. When we isolate the variable, we get (y-9)^(1/4) = -4. However, taking the fourth root of a number will always result in a real number, and since there is no real number that, when raised to the fourth power, equals -4, there are no real roots for this equation. Therefore, there is no solution.

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