Using the properties of exponents, which expression is equivalent to x16⋅x16

Question

(1 point)
Responses

x−−√
square root of x

x−−√36
36th root of x

x−−√3
cube root of x

1x√3
the fraction with numerator 1 and denominator cube root of x
Question 2
Using the properties of exponents, which expression is equivalent to xx34
(1 point)
Responses

x−−√4
4th root of x

x4
x to the 4th power

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

x
x
Question 3
Solve the following equation with a rational exponent: 2x32=16
(1 point)
Responses

4
4

8
8

no solution
no solution

64
64
Question 4
Solve the following equation with a rational exponent: 5x34=80
(1 point)
Responses

8
8

16
16

40.32
40.32

12
12
Question 5
Solve the following equation with a rational exponent: x23=64
(1 point)
Responses

32
32

16
16

512
512

8
8
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Answer 1

Let's go through each question one by one.

Question 1:

Using the properties of exponents, which expression is equivalent to \( x^{16} \cdot x^{16} \)?

Using the property of exponents that states \( a^m \cdot a^n = a^{m+n} \), \[ x^{16} \cdot x^{16} = x^{16+16} = x^{32}. \] Now, let's check the responses given:

• \( x^{-\sqrt{}} \) (seems unclear)
• \( x^{-\sqrt{36}} \)
• \( x^{-\sqrt{3}} \)
• \( \frac{1}{x^{\sqrt{3}}} \)

None of these options simplify to \( x^{32} \). Therefore, we need to revisit the question and responses as they seem unclear or incorrectly represented.

Question 2:

Using the properties of exponents, which expression is equivalent to \( x^{x^{34}} \)?

If we consider the interpretation from the inputs, it seems there was confusion in matching this exponent. In terms of simplification, we should be considering the following: \[ x^{34} \text{ is itself,} \] and the only option from the responses that looks like it represents a power of \( x \) appropriately is:

• \( x^4 \).

Thus the best fit here appears to be:

• \( x^4 \).

Question 3:

Solve the following equation with a rational exponent: \( 2x^{3/2} = 16 \)

First, divide both sides by 2: \[ x^{3/2} = 8. \] Now, raise both sides to the power of \( \frac{2}{3} \): \[ x = 8^{2/3}. \] Calculating \( 8^{2/3} \): \[ 8^{1/3} = 2 \implies 8^{2/3} = 2^2 = 4. \] Thus, the solution is:

• 4.

Question 4:

Solve the following equation with a rational exponent: \( 5x^{3/4} = 80 \)

First, divide both sides by 5: \[ x^{3/4} = 16. \] Now, raise both sides to the power of \( \frac{4}{3} \): \[ x = 16^{4/3}. \] Calculating \( 16^{4/3} \): \[ 16^{1/3} = 2 \implies 16^{4/3} = 2^4 = 16. \] Thus, the solution is:

• 16.

Question 5:

Solve the following equation with a rational exponent: \( x^{2/3} = 64 \)

Raise both sides to the power of \( \frac{3}{2} \): \[ x = 64^{3/2}. \] Calculating \( 64^{3/2} \): \[ 64^{1/2} = 8 \implies 64^{3/2} = 8^3 = 512. \] Thus, the solution is:

• 512.

Please verify responses with expected findings since there’s an ambiguity in certain definitions in the response layouts provided.