Question

1. According to the exponent rules, what is the rational exponent in the equivalent expression to 5–√3
? 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** \(-\frac{3}{2}\).

---

**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:

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

---

**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** \( 5 \).

---

**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**.

---

**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**.

---

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