To simplify the expression [tex]\(\sqrt{20 y^8}\)[/tex] where [tex]\(y\)[/tex] is a positive real number, we can follow these steps:
1. Factorize the expression inside the square root:
The term inside the square root, [tex]\(20 y^8\)[/tex], can be broken down into its prime factors and powers:
[tex]\[
20 = 4 \times 5
\][/tex]
Therefore,
[tex]\[
20 y^8 = 4 \times 5 \times y^8
\][/tex]
2. Separate the factors under the square root:
Use the property of square roots that [tex]\(\sqrt{a \times b \times c} = \sqrt{a} \times \sqrt{b} \times \sqrt{c}\)[/tex]:
[tex]\[
\sqrt{20 y^8} = \sqrt{4 \times 5 \times y^8} = \sqrt{4} \times \sqrt{5} \times \sqrt{y^8}
\][/tex]
3. Simplify the square roots of the separate factors:
- The square root of [tex]\(4\)[/tex] is [tex]\(2\)[/tex]:
[tex]\[
\sqrt{4} = 2
\][/tex]
- The square root of [tex]\(5\)[/tex] remains [tex]\(\sqrt{5}\)[/tex] because 5 is a prime number and does not have a rational square root:
[tex]\[
\sqrt{5} = \sqrt{5}
\][/tex]
- For the variable part [tex]\(y^8\)[/tex], we use the rule that [tex]\(\sqrt{y^n} = y^{n/2}\)[/tex]:
[tex]\[
\sqrt{y^8} = y^{8/2} = y^4
\][/tex]
4. Combine the simplified results:
Now, we can combine all the simplified parts back together:
[tex]\[
\sqrt{20 y^8} = 2 \times \sqrt{5} \times y^4
\][/tex]
Therefore, the simplified form of the given expression [tex]\(\sqrt{20 y^8}\)[/tex] is:
[tex]\[
2 \sqrt{5} y^4
\][/tex]
