Let's go through each question step by step.
Question 1:
Using the properties of exponents, you want to simplify \( x^{16} \cdot x^{16} \).
Using the property \( a^m \cdot a^n = a^{m+n} \): \[ x^{16} \cdot x^{16} = x^{16+16} = x^{32} \]
None of the provided responses seem to match \( x^{32} \). Please double-check the options listed.
Question 2:
You want to simplify \( x^{x^{3/4}} \) or more likely \( x \cdot x^{3/4} \).
Using the property again \( a^m \cdot a^n = a^{m+n} \): \[ x \cdot x^{3/4} = x^{1 + 3/4} = x^{7/4} \]
None of the provided options appear to match \( x^{7/4} \).
Question 3:
Solve the equation \( 2x^{3/2} = 16 \).
Divide both sides by 2: \[ x^{3/2} = 8 \]
Raise both sides to the power of \( \frac{2}{3} \): \[ x = 8^{2/3} \] Find \( 8^{2/3} \): \[ 8^{2/3} = (2^3)^{2/3} = 2^{3 \cdot (2/3)} = 2^2 = 4 \]
So the answer is 4.
Question 4:
Solve the equation \( 5x^{4/3} = 80 \).
Divide both sides by 5: \[ x^{4/3} = 16 \]
Raise both sides to the power of \( \frac{3}{4} \): \[ x = 16^{3/4} \] Find \( 16^{3/4} \): \[ 16^{3/4} = (2^4)^{3/4} = 2^{4 \cdot (3/4)} = 2^3 = 8 \]
So the answer is 8.
Question 5:
Solve the equation \( x^{2/3} = 64 \).
Raise both sides to the power of \( \frac{3}{2} \): \[ x = 64^{3/2} \] Find \( 64^{3/2} \): \[ 64^{3/2} = (8^2)^{3/2} = 8^{2 \cdot (3/2)} = 8^3 = 512 \]
So the answer is 512.
Summary of Answers:
- Review options for \( x^{32} \)
- Review options for \( x^{7/4} \)
- 4
- 8
- 512