Question
1. Using the properties of exponents, which expression is equivalent to x1/6⋅x1/6
(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
(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
Answers
GPT-4o mini
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 \( x \)
---
### 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 \( x \)
---
### 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 \( 8^{2/3} \):
\[
8^{1/3} = 2 \quad \text{(since the cube root of 8 is 2)}
\]
\[
2^2 = 4
\]
**Correct response:** 4
---
### 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 \( 16^{3/4} \):
\[
16^{1/4} = 2 \quad \text{(since the 4th root of 16 is 2)}
\]
\[
2^3 = 8
\]
**Correct response:** 8
---
### Question 5
**Solve the equation \( x^{2/3} = 64 \)**.
Raise both sides to the power of \( \frac{3}{2} \):
\[
x = 64^{3/2}
\]
Finding \( 64^{3/2} \):
\[
64^{1/2} = 8 \quad \text{(since the square root of 64 is 8)}
\]
\[
8^3 = 512
\]
**Correct response:** 512
---
Here are the final answers for each question:
1. cube root of \( x \)
2. 4th root of \( x \)
3. 4
4. 8
5. 512
### 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 \( x \)
---
### 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 \( x \)
---
### 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 \( 8^{2/3} \):
\[
8^{1/3} = 2 \quad \text{(since the cube root of 8 is 2)}
\]
\[
2^2 = 4
\]
**Correct response:** 4
---
### 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 \( 16^{3/4} \):
\[
16^{1/4} = 2 \quad \text{(since the 4th root of 16 is 2)}
\]
\[
2^3 = 8
\]
**Correct response:** 8
---
### Question 5
**Solve the equation \( x^{2/3} = 64 \)**.
Raise both sides to the power of \( \frac{3}{2} \):
\[
x = 64^{3/2}
\]
Finding \( 64^{3/2} \):
\[
64^{1/2} = 8 \quad \text{(since the square root of 64 is 8)}
\]
\[
8^3 = 512
\]
**Correct response:** 512
---
Here are the final answers for each question:
1. cube root of \( x \)
2. 4th root of \( x \)
3. 4
4. 8
5. 512