To simplify the expression [tex]\(\sqrt{5} \times \sqrt{5} \times \sqrt{5}\)[/tex] and express it in index form, follow these steps:
1. Understand the Expression: The expression consists of the product of three identical terms, each of which is [tex]\(\sqrt{5}\)[/tex].
2. Use the Property of Square Roots: Recall that the square root can be expressed as an exponent. Specifically, [tex]\(\sqrt{5} = 5^{1/2}\)[/tex]. Thus, we can rewrite [tex]\(\sqrt{5} \times \sqrt{5} \times \sqrt{5}\)[/tex] as:
[tex]\[
(5^{1/2}) \times (5^{1/2}) \times (5^{1/2})
\][/tex]
3. Apply the Exponent Addition Rule: When multiplying exponents with the same base, you add the exponents. Therefore:
[tex]\[
5^{1/2} \times 5^{1/2} \times 5^{1/2} = 5^{(1/2) + (1/2) + (1/2)}
\][/tex]
4. Simplify the Exponent: Add the exponents together:
[tex]\[
(1/2) + (1/2) + (1/2) = 3 \times (1/2) = 3/2
\][/tex]
So, the expression simplifies to:
[tex]\[
5^{3/2}
\][/tex]
5. Express the Result: Notice that [tex]\(5^{3/2}\)[/tex] can be broken down further into a more recognizable form. Specifically,
[tex]\[
5^{3/2} = (5^1) \times (5^{1/2})
\][/tex]
6. Combine the Expressions: We know from earlier that [tex]\(5^{1/2}\)[/tex] is [tex]\(\sqrt{5}\)[/tex], so:
[tex]\[
5^{3/2} = 5 \times 5^{1/2} = 5 \times \sqrt{5}
\][/tex]
Therefore, the simplified and indexed form of the expression [tex]\(\sqrt{5} \times \sqrt{5} \times \sqrt{5}\)[/tex] is:
[tex]\[
5\sqrt{5}
\][/tex]
