To solve the expression [tex]\(\sqrt{75 y^7}\)[/tex], we can follow these detailed steps:
1. Rewrite the Radical Expression:
The square root expression [tex]\(\sqrt{75 y^7}\)[/tex] can be rewritten using the properties of square roots, specifically that [tex]\(\sqrt{a \cdot b} = \sqrt{a} \cdot \sqrt{b}\)[/tex].
2. Factor the Radicand:
We aim to simplify the radicand, which is [tex]\(75 y^7\)[/tex]. Notice that [tex]\(75\)[/tex] can be factored into [tex]\(3 \cdot 25\)[/tex], and [tex]\(25\)[/tex] is a perfect square:
[tex]\[
75 = 3 \cdot 25 = 3 \cdot 5^2
\][/tex]
So, we now have:
[tex]\[
75 y^7 = 3 \cdot 5^2 \cdot y^7
\][/tex]
3. Separate and Simplify the Square Root:
Next, we apply the square root to each part of the product inside the radical:
[tex]\[
\sqrt{75 y^7} = \sqrt{3 \cdot 5^2 \cdot y^7}
\][/tex]
By breaking it up, we get:
[tex]\[
\sqrt{75 y^7} = \sqrt{3} \cdot \sqrt{5^2} \cdot \sqrt{y^7}
\][/tex]
4. Simplify Individual Components:
We know that [tex]\(\sqrt{5^2} = 5\)[/tex]. We'll now simplify [tex]\(\sqrt{y^7}\)[/tex]:
The exponent [tex]\(7\)[/tex] can be split as [tex]\(6 + 1\)[/tex]:
[tex]\[
y^7 = y^{6+1} = y^6 \cdot y = (y^3)^2 \cdot y
\][/tex]
Take the square root of each part:
[tex]\[
\sqrt{y^7} = \sqrt{(y^3)^2 \cdot y} = \sqrt{(y^3)^2} \cdot \sqrt{y} = y^3 \cdot \sqrt{y}
\][/tex]
5. Combine the Simplified Terms:
Now, combine all the simplified parts:
[tex]\[
\sqrt{75 y^7} = \sqrt{3} \cdot 5 \cdot y^3 \cdot \sqrt{y}
\][/tex]
Grouping the terms in a more simplified form, we obtain:
[tex]\[
5 y^3 \sqrt{3y}
\][/tex]
Therefore, the simplified form of [tex]\(\sqrt{75 y^7}\)[/tex] is:
[tex]\[
5 y^3 \sqrt{3y}
\][/tex]
