Question

Radicals and rationals exponents practice use the properties of exponents to generate the equivalent expression for three square root 8 over square root 2 find the cube root of 8 then rewrite rational exponents as radicals and simple

Answer 1

To simplify the expression \( \frac{\sqrt{8}}{\sqrt{2}} \) using properties of exponents, we'll break down the steps.

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

2. **Express \(8\) in terms of \(2\)**:
Since \(8 = 2^3\), we can replace \(8\) in our expression:
\[
8^{1/2} = (2^3)^{1/2} = 2^{3/2}
\]

3. **Substitute**:
Now substitute \(8^{1/2}\) into our expression:
\[
\frac{8^{1/2}}{2^{1/2}} = \frac{2^{3/2}}{2^{1/2}}
\]

4. **Use the properties of exponents** to simplify:
When dividing exponents with the same base, we subtract the exponents:
\[
2^{3/2 - 1/2} = 2^{(3/2 - 1/2)} = 2^{2/2} = 2^1 = 2
\]

So,
\[
\frac{\sqrt{8}}{\sqrt{2}} = 2
\]

5. **Finding the cube root of \(8\)**:
The cube root of \(8\) is:
\[
\sqrt[3]{8} = 2
\]

6. **Rewriting rational exponents as radicals**:
To rewrite a rational exponent \(x^{m/n}\) as a radical, it can be represented as:
\[
x^{m/n} = \sqrt[n]{x^m}
\]
For example:
- \(8^{1/3} = \sqrt[3]{8}\)

Both rational exponent formats and their corresponding radical forms are equivalent, and we have simplified the main expression \( \frac{\sqrt{8}}{\sqrt{2}} \) to \(2\) and found the cube root of \(8\) also to be \(2\).