1. According to the exponent rules, what is the rational exponent in the equivalent expression to 5–√3

Question

? Enter your response in fraction form. (Hint: what should go in the yellow box)(1 point)
The rational exponent is

Question 2
2. Use the properties of exponents to generate an equivalent expression for 3√8/√2
(1 point)
3√8/√2=


Question 3
3. Use the Quotient Property and Negative Exponent Property to generate an equivalent expression for 49√/7^−3
. What will the rational exponent be for the new expression? (Hint: what should go in the yellow box)(1 point)
The rational exponent for the new expression is

Question 4
4. Consider the equation 4^3⋅(3√64)^2=4^6
. Is this equation true or false? (1 point)
Responses

True
True

False
False
Question 5
5. Consider the equation: (√25)^4⋅5^6=5^8
. Is this equation true or false?(1 point)
Responses

True
True

False

Answer 1

Let's go through each question one by one.

Question 1: The expression \( 5^{-\sqrt{3}} \) can be rewritten using the rule of rational exponents. The negative exponent indicates a reciprocal, and the square root can be expressed as an exponent of \( \frac{1}{2} \). Thus:

\[ -\sqrt{3} = -3^{\frac{1}{2}} = -\frac{3}{2} \]

So the rational exponent is:

The rational exponent is \(-\frac{3}{2}\).

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Question 2: For \( \frac{3\sqrt{8}}{\sqrt{2}} \), we can rewrite it using exponents:

\[ 3\sqrt{8} = 3 \cdot 8^{\frac{1}{2}} = 3 \cdot (2^3)^{\frac{1}{2}} = 3 \cdot 2^{\frac{3}{2}} \] \[ \sqrt{2} = 2^{\frac{1}{2}} \]

Now substituting these into the expression:

\[ \frac{3 \cdot 2^{\frac{3}{2}}}{2^{\frac{1}{2}}} = 3 \cdot 2^{\frac{3}{2} - \frac{1}{2}} = 3 \cdot 2^{\frac{2}{2}} = 3 \cdot 2^1 = 3 \cdot 2 = 6 \]

Thus:

\( 3\sqrt{8}/\sqrt{2} = 6 \).

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Question 3: For \( \frac{49\sqrt{}}{7^{-3}} \), we first rewrite \( 49 \) and the exponent properties:

• Note that \( 49 = 7^2 \).
• Thus, we can write:

\[ \frac{49}{7^{-3}} = \frac{7^2}{7^{-3}} = 7^{2 - (-3)} = 7^{2 + 3} = 7^5 \]

The rational exponent in the new expression \( 7^5 \) is simply:

The rational exponent for the new expression is \( 5 \).

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Question 4: To determine if \( 4^3 \cdot (3\sqrt{64})^2 = 4^6 \) is true or false:

• First, calculate \( (3\sqrt{64})^2 \): \[ 3\sqrt{64} = 3 \cdot 8 = 24 \Rightarrow (24)^2 = 576 \]

Now, calculate \( 4^3 \) and \( 4^6 \):

• \( 4^3 = 64 \), and \( 4^6 = 4096 \).

Now, check the equation: \[ 64 \cdot 576 \neq 4096 \]

So the equation is False.

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Question 5: For the equation \( (\sqrt{25})^4 \cdot 5^6 = 5^8 \):

First, calculate \( \sqrt{25} \):

• \( \sqrt{25} = 5 \).

Now, evaluating \( (\sqrt{25})^4 \): \[ (5)^4 = 625, ; 5^6 = 15625. \]

Now check: \[ 625 \cdot 15625 \neq 5^8 (which equals 390625). \]

Now, notice:

• \( (5)^4 \cdot (5)^6 = (5)^{4+6} = 5^{10} \), which is not equal to \( 5^8 \).

So the equation is False.

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To summarize:

• Question 1: \(-\frac{3}{2}\)
• Question 2: \(6\)
• Question 3: \(5\)
• Question 4: False
• Question 5: False