Question

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

Answer 1

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

Answer 2

Write the answer in word form

Answer 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?

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

Answer 4

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 \( 2 \).

Answer 5

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:
\[
(3^{\sqrt{2}})^2 \cdot \frac{1}{\sqrt{2}} \quad \text{or} \quad \frac{(3^{\sqrt{2}})^2}{\sqrt{2}}
\]

Answer 6

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."

Answer 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."