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:
- cube root of \( x \)
- 4th root of \( x \)
- 4
- 8
- 512