According to exponent rules, what is the rational exponent in the equivilent expression to the square root of five cubed

The square root of \( 5^3 \) can be expressed using rational exponents. The square root can be represented as an exponent of \( \frac{1}{2} \). Therefore, we have:

\[
\sqrt{5^3} = (5^3)^{\frac{1}{2}}
\]

Using the exponent multiplication rule \((a^m)^n = a^{m \cdot n}\), we can simplify this expression:

\[
(5^3)^{\frac{1}{2}} = 5^{3 \cdot \frac{1}{2}} = 5^{\frac{3}{2}}
\]

Thus, the rational exponent in the equivalent expression for the square root of five cubed is \(\frac{3}{2}\).

Say it in word form

The rational exponent in the equivalent expression for the square root of five cubed is three over two.

Wrong

I apologize for any confusion. The correct interpretation of the square root of five cubed in word form is:

"The square root of five cubed has a rational exponent of one-half times three, which simplifies to three over two."

If this does not address your question correctly, please clarify further, and I'll be happy to assist!

The index of the radicand gives key information when generating a rational exponent.

You're correct! The index of the radicand in a radical expression is essential for generating the corresponding rational exponent.

In the case of the square root of \(5^3\), the index of the square root is 2 (since square roots are traditionally understood to have an index of 2). Therefore, when converting the square root to its rational exponent form, you take the exponent of the radicand (which is 3) and divide it by the index (which is 2).

This means:

\[
\sqrt{5^3} = (5^3)^{\frac{1}{2}} = 5^{\frac{3}{2}}
\]

So, the key takeaway is that the index informs you that you need to divide the exponent by 2 when converting to a rational exponent. In word form, you would say:

"The rational exponent for the square root of five cubed is three over two, where the index of the square root is two."

Thank you for your patience, and I hope this clarification helps!

Use the properties of exponents to generate an equivalent expression for 8√32√

Write your answer in word form

To generate an equivalent expression for \(8 \sqrt{32}\), we first rewrite the square root in terms of rational exponents. The square root of \(32\) can be expressed as \(32^{1/2}\). Then, we can rewrite the expression:

\[
8 \sqrt{32} = 8 \times 32^{1/2}
\]

Next, we note that \(8\) can be expressed as \(8 = 2^3\), and \(32\) can be expressed as \(32 = 2^5\). Therefore, substituting these into the expression gives:

\[
8 \sqrt{32} = 2^3 \times (2^5)^{1/2}
\]

Using the property of exponents \((a^m)^n = a^{m \cdot n}\), we have:

\[
(2^5)^{1/2} = 2^{5/2}
\]

Now combining the exponents, we can write:

\[
8 \sqrt{32} = 2^3 \times 2^{5/2} = 2^{3 + 5/2}
\]

To add the exponents, we need a common denominator. The number \(3\) can be expressed as \(\frac{6}{2}\):

\[
3 + \frac{5}{2} = \frac{6}{2} + \frac{5}{2} = \frac{11}{2}
\]

Thus, we can simplify the expression further:

\[
8 \sqrt{32} = 2^{11/2}
\]

In word form, this would be expressed as:

"Eighty multiplied by the square root of thirty-two is equivalent to two raised to the eleven over two power."