rewrite rational exponents as radicals in the cubed root of 8 then simplify

To rewrite the rational exponent of the cube root of 8 and simplify, follow these steps:

1. The cube root of 8 can be written using rational exponents as \( 8^{1/3} \).
2. Now, express 8 as \( 2^3 \):
\[
8^{1/3} = (2^3)^{1/3}
\]
3. Apply the power of a power property, which states that \( (a^m)^n = a^{m \cdot n} \):
\[
(2^3)^{1/3} = 2^{3 \cdot \frac{1}{3}} = 2^1 = 2
\]

Thus, the simplified form of the cube root of 8 is \( 2 \).

Use the properties of exponents to generate an equivilent expression for 3^square root for 8/square root of 2

To generate an equivalent expression for \( \frac{3^{\sqrt{8}}}{\sqrt{2}} \) using properties of exponents, follow these steps:

1. **Rewrite the square roots as exponents**:
\[
\sqrt{8} = 8^{1/2} \quad \text{and} \quad \sqrt{2} = 2^{1/2}
\]
So, we can rewrite the expression:
\[
\frac{3^{\sqrt{8}}}{\sqrt{2}} = \frac{3^{8^{1/2}}}{2^{1/2}}
\]

2. **Simplify \( \sqrt{8} \)**:
\[
\sqrt{8} = \sqrt{4 \times 2} = \sqrt{4} \cdot \sqrt{2} = 2\sqrt{2}
\]
Thus, we rewrite \( 3^{\sqrt{8}} \) as:
\[
3^{\sqrt{8}} = 3^{2\sqrt{2}} = (3^{\sqrt{2}})^2
\]

3. **Combine the expression**:
Now we can combine both parts:
\[
\frac{(3^{\sqrt{2}})^2}{2^{1/2}}
\]

Putting it all together, we end up with the equivalent expression:
\[
\frac{(3^{\sqrt{2}})^2}{2^{1/2}} = (3^{\sqrt{2}})^2 \cdot 2^{-1/2}
\]

Thus, the final expression can be written in an alternative simplified form:
\[
(3^{\sqrt{2}})^2 \cdot \frac{1}{\sqrt{2}} \quad \text{or} \quad \frac{(3^{\sqrt{2}})^2}{\sqrt{2}}
\]

Write the answer in word form

The equivalent expression for three raised to the square root of eight divided by the square root of two can be written in word form as:

"The square of three raised to the square root of two divided by the square root of two."

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?

Write the answer in word from and make sure its an exponent for 7

To simplify the expression \( 49\sqrt{7^{-3}} \) using the Quotient Property and the Negative Exponent Property, let's follow these steps:

1. **Rewrite the square root using rational exponents**:
\[
\sqrt{7^{-3}} = (7^{-3})^{1/2} = 7^{-3/2}
\]

2. **Substituting back into the expression**:
\[
49\sqrt{7^{-3}} = 49 \cdot 7^{-3/2}
\]

3. **Expressing 49 as a power of 7**:
Since \( 49 = 7^2 \), we can rewrite the entire expression:
\[
7^2 \cdot 7^{-3/2}
\]

4. **Using the Quotient Property** (which states that \( a^m \cdot a^n = a^{m+n} \)):
\[
7^{2 - 3/2}
\]
Now, convert \( 2 \) to have a common denominator with \( 3/2 \):
\[
2 = \frac{4}{2}
\]
So we add the exponents:
\[
7^{\frac{4}{2} - \frac{3}{2}} = 7^{\frac{1}{2}}
\]

### Final Expression and Word Form
The final equivalent expression is \( 7^{\frac{1}{2}} \).

In word form, this can be stated as:

"The square root of seven."