Question

Which is an equivalent expression for 5^1/2/^3 square root 125

Answer 1

It’s 5^1/2 over ^3 square root 125

Answer 2

Options
5^0
1
square root 5 over 5
1/5

Answer 3

To simplify the expression \( 5^{1/2} \cdot \sqrt{125} \), let's break it down step by step.

1. **Calculate \( \sqrt{125} \)**:
\[
125 = 25 \times 5 = 5^2 \times 5 = 5^3
\]
Therefore,
\[
\sqrt{125} = \sqrt{5^3} = 5^{3/2}
\]

2. **Combine the expressions**:
We now have:
\[
5^{1/2} \cdot \sqrt{125} = 5^{1/2} \cdot 5^{3/2}
\]

3. **Use properties of exponents**:
When multiplying like bases, we add the exponents:
\[
5^{1/2} \cdot 5^{3/2} = 5^{(1/2 + 3/2)} = 5^{4/2} = 5^2
\]

4. **Final result**:
Therefore,
\[
5^{1/2} \cdot \sqrt{125} = 5^2 = 25
\]

So, an equivalent expression for \( 5^{1/2} \cdot \sqrt{125} \) is \( 25 \).

Answer 4

Thank you for the clarification! It appears you're referring to the expression \( \frac{5^{1/2}}{\sqrt[3]{125}} \). Let’s simplify this expression step by step.

1. **Calculate \( \sqrt[3]{125} \)**:
\[
125 = 5^3
\]
Thus,
\[
\sqrt[3]{125} = \sqrt[3]{5^3} = 5^{3/3} = 5^1 = 5
\]

2. **Rewrite the expression**:
Now substituting back into the expression, we have:
\[
\frac{5^{1/2}}{\sqrt[3]{125}} = \frac{5^{1/2}}{5}
\]

3. **Simplify the expression**:
When we divide like bases, we subtract the exponents:
\[
\frac{5^{1/2}}{5^1} = 5^{1/2 - 1} = 5^{1/2 - 2/2} = 5^{-1/2}
\]

4. **Final result**:
Therefore, \(\frac{5^{1/2}}{\sqrt[3]{125}} = 5^{-1/2}\), which can also be expressed as:
\[
\frac{1}{\sqrt{5}}
\]

So the equivalent expression for \( \frac{5^{1/2}}{\sqrt[3]{125}} \) is \( 5^{-1/2} \) or \( \frac{1}{\sqrt{5}} \).

Answer 5

Given the expression \( \frac{5^{1/2}}{\sqrt[3]{125}} \) that we previously simplified to \( 5^{-1/2} \), we can find the value among the provided options:

1. **Recall that**:
\[
5^{-1/2} = \frac{1}{5^{1/2}} = \frac{1}{\sqrt{5}}.
\]

Now let’s analyze the given options:

1. \( 5^0 = 1 \)
2. \( 1 \)
3. \( \frac{\sqrt{5}}{5} \)
4. \( \frac{1}{5} \)

Since we simplified \( \frac{5^{1/2}}{\sqrt[3]{125}} \) to \( \frac{1}{\sqrt{5}} \), we can also express \( \frac{1}{\sqrt{5}} \) in terms of the given options.

To rewrite \( \frac{1}{\sqrt{5}} \):
- It can be expressed as \( \frac{\sqrt{5}}{5} \) since \( \frac{1}{\sqrt{5}} = \frac{\sqrt{5}}{5} \) upon multiplying the numerator and the denominator by \( \sqrt{5} \).

Thus, the equivalent expression from the provided options is:
**\( \frac{\sqrt{5}}{5} \)**.