Question

Which of the following is equivalent to
4
√
y
3/
√
y
5
?(1 point)
Responses

y
7
4
y superscript Start Fraction 7 over 4 End Fraction baseline

y
13
4
y superscript Start Fraction 13 over 4 End Fraction baseline

y
14
15
y superscript Start Fraction 14 over 15 End Fraction baseline

1
y
7
4

Answer 1

To simplify the expression \( \frac{4 \sqrt{y^3}}{\sqrt{y^5}} \), we can follow these steps:

1. Rewrite the square roots in terms of exponents.
\[
\sqrt{y^3} = y^{3/2} \quad \text{and} \quad \sqrt{y^5} = y^{5/2}
\]

2. Substitute these back into the expression:
\[
\frac{4 \cdot y^{3/2}}{y^{5/2}}
\]

3. Use the property of exponents to simplify:
\[
\frac{y^{3/2}}{y^{5/2}} = y^{3/2 - 5/2} = y^{-2/2} = y^{-1}
\]

4. So, the expression becomes:
\[
4y^{-1} = \frac{4}{y}
\]

Now, let's compare this to the provided options. The answer in simplified form is

\[
\frac{4}{y} = 4y^{-1}
\]

None of the responses given match this simplified form directly, but the closest option is:

\[
1/y^{7/4}
\]
If options were more likely to represent radical expressions, \( y^{-1} \) can typically represent a simplified form in many contexts, yet is still distinct from \( \frac{4}{y} \).

If there are more specific options or if I misunderstood the notation provided, please clarify! Based on this setup, none is equivalent to \( \frac{4}{y} \) among the given options completely.