1. Using the properties of exponents, which expression is equivalent to x1/6⋅x1/6

Question

(1 point)
Responses

x−−√3
cube root of x

x−−√
square root of x

x−−√36
36th root of x

1x√3
the fraction with numerator 1 and denominator cube root of x
Question 2
2. Using the properties of exponents, which expression is equivalent to x/x3/4
(1 point)
Responses

x
x

1x√4
the fraction with numerator 1 and denominator 4th root of x

x−−√4
4th root of x

x4
x to the 4th power
Question 3
3. Solve the following equation with a rational exponent: 2x3/2=16
(1 point)
Responses

no solution
no solution

64
64

8
8

4
4
Question 4
4. Solve the following equation with a rational exponent: 5x4/3=80
(1 point)
Responses

12
12

40.32
40.32

8
8

16
16
Question 5
5. Solve the following equation with a rational exponent: x2/3=64
(1 point)
Responses

32
32

8
8

512
512

16

Answer 1

Let's go through each of the questions step by step.

Question 1

Using the properties of exponents, which expression is equivalent to $x^{1/6}\cdot x^{1/6}$?

Using the law of exponents, $a^m\cdot a^n=a^{m+n}$: $$x^{1/6}\cdot x^{1/6}=x^{(1/6+1/6)}=x^{2/6}=x^{1/3}$$

The expression $x^{1/3}$ is the same as the cube root of $x$.

Correct response: cube root of $x$

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Question 2

Using the properties of exponents, which expression is equivalent to $\frac{x}{x^{3/4}}$?

Using the law of exponents, $\frac{a^m}{a^n}=a^{m-n}$: $$\frac{x}{x^{3/4}}=x^{1-3/4}=x^{1/4}$$

The expression $x^{1/4}$ is the same as the 4th root of $x$.

Correct response: 4th root of $x$

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Question 3

Solve the equation $2x^{3/2}=16$.

First, divide both sides by 2: $$x^{3/2}=8$$

Next, raise both sides to the power of $\frac{2}{3}$: $$x=8^{2/3}$$

Finding $8^{2/3}$: $$8^{1/3}=2\text{(since the cube root of 8 is 2)}$$ $$2^2=4$$

Correct response: 4

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Question 4

Solve the equation $5x^{4/3}=80$.

First, divide both sides by 5: $$x^{4/3}=16$$

Next, raise both sides to the power of $\frac{3}{4}$: $$x={16}^{3/4}$$

Finding ${16}^{3/4}$: $${16}^{1/4}=2\text{(since the 4th root of 16 is 2)}$$ $$2^3=8$$

Correct response: 8

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Question 5

Solve the equation $x^{2/3}=64$.

Raise both sides to the power of $\frac{3}{2}$: $$x={64}^{3/2}$$

Finding ${64}^{3/2}$: $${64}^{1/2}=8\text{(since the square root of 64 is 8)}$$ $$8^3=512$$

Correct response: 512

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Here are the final answers for each question:

1. cube root of $x$
2. 4th root of $x$
3. 4
4. 8
5. 512