1.

Question

Charles, Zayeer, and Kali are trying to simplify 1000^2/3
.

Charles says the correct simplification is 10 because 1000=10⋅10⋅10
and 100023=10
.

Zayeer says that the correct simplification is 1000 because 1000=10⋅10⋅10→100023=10⋅10⋅10=1000
.

Kali says the correct simplification is 100 because 1000=10⋅10⋅10→100023=10⋅10=100
.

Who has the correct value?

(2 points)
Responses

none of them
none of them

Zayeer
Zayeer

Charles
Charles

Kali
Kali
Question 2
2.

Simplify 243^3/5

(2 points)
243^3/5
=
Question 3
3.

Write the radical form of the expression 25^1/4

(2 points)
25^1/4 =

Question 4
4.

What is the simplest form of 25^1/2
?

(2 points)
Responses

25−−√
square root of 25

5
5

12.5
12.5

25−−√12
1 halfth root of 25
Question 5
5.

Rewrite the following expressions:

27^1/2⋅27^1/4

(2 points)
The combined expression is 27 to the power of
(write as a fraction)
Question 6
6.

What is an equivalent expression for −(b^1/4)^12
?

(2 points)
Responses

−b−3
negative b to the negative 3 power

−b3
negative b cubed

1b3
the fraction with numerator 1 and denominator b cubed

−1b3
negative the fraction with numerator 1 and denominator b cubed
Question 7
7.

Find the value of x.

−x^3/2=−27

(2 points)
x=

Question 8
8.

Solve for x.

27x^3/4−1=26

(2 points)
x=

Question 9
9.

Using exponent properties, which of the following expressions is equivalent to (2–√)^3⋅2^−1/2
?

(2 points)
Responses

212
2 raised to the 1 half power

2–√
square root of 2

2
2

22
2 squared
Question 10
10.

When applying the exponent properties, what is the equivalent expression to square root of 216 / 27^1/3
?

(2 points)
the equivalent expression is:
Question 11
11.

The equation (8–√4)^−24⋅(8–√)^16=8^−48
is false. Select the responses that explain what was done incorrectly.

(4 points)
The equation is false because the exponents should be (added, subtracted, multiplied, or divided)
rather than (added, subtracted, multiplied, or divided)
Question 12
12.

Simplify the expression

7^−4⋅7^3/49√

(2 points)
The simplified expression is
(do not include negative exponents) (Answer should be a simplified fraction)
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Answer 1

Let's address each question step by step:

Question 1:

1. Correct simplification of \( \frac{1000^2}{3} \): \[ 1000 = 10^3 \] \[ 1000^{2/3} = (10^3)^{2/3} = 10^{3 \cdot \frac{2}{3}} = 10^2 = 100 \] Kali is correct.

Question 2:

2. Simplify \( 243^{3/5} \): \[ 243 = 3^5 \quad (\text{since } 3^5 = 243) \] \[ 243^{3/5} = (3^5)^{3/5} = 3^{(5 \cdot \frac{3}{5})} = 3^3 = 27 \] So, \( 243^{3/5} = 27 \).

Question 3:

3. Write the radical form of the expression \( 25^{1/4} \): \[ 25^{1/4} = \sqrt[4]{25} \]

Question 4:

4. Simplest form of \( 25^{1/2} \): \[ 25^{1/2} = \sqrt{25} = 5 \]

Question 5:

5. Rewrite the expression \( 27^{1/2} \cdot 27^{1/4} \): \[ 27^{1/2} \cdot 27^{1/4} = 27^{1/2 + 1/4} = 27^{2/4 + 1/4} = 27^{3/4} \]

Question 6:

6. Equivalent expression for \( -\left(b^{1/4}\right)^{12} \): \[ -\left(b^{1/4}\right)^{12} = -b^{(1/4) \cdot 12} = -b^3 \] Therefore, the answer is: −b^3.

Question 7:

7. Find the value of \( x \) in \( -x^{3/2} = -27 \): Divide both sides by -1: \[ x^{3/2} = 27 \] To solve for \( x \), raise both sides to the power of \( \frac{2}{3} \): \[ x = 27^{\frac{2}{3}} = (3^3)^{\frac{2}{3}} = 3^2 = 9 \] Hence, \( x = 9 \).

Question 8:

8. Solve for \( x \) in \( 27x^{3/4} - 1 = 26 \): \[ 27x^{3/4} = 27 \] Divide by 27: \[ x^{3/4} = 1 \] Raise both sides to \( \frac{4}{3} \): \[ x = 1^{\frac{4}{3}} = 1 \]

Question 9:

9. Using exponent properties, which is equivalent to \( (2^{-\frac{1}{2}})^3 \cdot 2^{-\frac{1}{2}} \): \[ 2^{-\frac{3}{2}} \cdot 2^{-\frac{1}{2}} = 2^{-\frac{3}{2} - \frac{1}{2}} = 2^{-2} = \frac{1}{4} \]

Question 10:

10. Find the equivalent expression for \( \frac{\sqrt{216}}{27^{1/3}} \): First simplify each part.

• \( \sqrt{216} = \sqrt{36 \cdot 6} = 6\sqrt{6} \)
• \( 27^{1/3} = 3 \)

Thus, \[ \frac{6\sqrt{6}}{3} = 2\sqrt{6} \]

Question 11:

11. For the equation \( (8^{-\sqrt{4}})^{-24} \cdot (8^{-\sqrt{3}})^{16} = 8^{-48} \):

• Notice that both terms on the left should use multiplication of exponents, leading to:
• The equation is false because the exponents should be added rather than multiplied.

Question 12:

12. Simplify \( 7^{-4} \cdot 7^{3}/\sqrt{49} \): \[ \sqrt{49} = 7 \quad \text{so} \quad \frac{7^{3}}{7} = 7^{2} \] Then, \[ 7^{-4} \cdot 7^{2} = 7^{-4 + 2} = 7^{-2} = \frac{1}{7^{2}} = \frac{1}{49} \]

Summary of Answers:

1. Kali
2. \( 27 \)
3. \( \sqrt[4]{25} \)
4. \( 5 \)
5. \( \frac{3}{4} \)
6. −b^3
7. \( 9 \)
8. \( 1 \)
9. \( 2^{-2} \) (or \( \frac{1}{4} \))
10. \( 2\sqrt{6} \)
11. The equation is false because the exponents should be added rather than multiplied.
12. \( \frac{1}{49} \)