To simplify the expression [tex]\(\sqrt{a^5 b^3 c d^8}\)[/tex], we'll follow these steps:
1. Separate the expression under the square root:
The square root of a product is the product of the square roots of each factor. So, we can write:
[tex]\[
\sqrt{a^5 b^3 c d^8} = \sqrt{a^5} \cdot \sqrt{b^3} \cdot \sqrt{c} \cdot \sqrt{d^8}
\][/tex]
2. Simplify each square root individually:
- For [tex]\(\sqrt{a^5}\)[/tex]:
[tex]\[
\sqrt{a^5} = \sqrt{a^4 \cdot a} = \sqrt{a^4} \cdot \sqrt{a}
\][/tex]
Since [tex]\(a^4\)[/tex] is a perfect square ([tex]\((a^2)^2\)[/tex]):
[tex]\[
\sqrt{a^4} = a^2
\][/tex]
Therefore:
[tex]\[
\sqrt{a^5} = a^2 \sqrt{a}
\][/tex]
- For [tex]\(\sqrt{b^3}\)[/tex]:
[tex]\[
\sqrt{b^3} = \sqrt{b^2 \cdot b} = \sqrt{b^2} \cdot \sqrt{b}
\][/tex]
Since [tex]\(b^2\)[/tex] is a perfect square ([tex]\(b^2 = (b)^2\)[/tex]):
[tex]\[
\sqrt{b^2} = b
\][/tex]
Therefore:
[tex]\[
\sqrt{b^3} = b \sqrt{b}
\][/tex]
- For [tex]\(\sqrt{c}\)[/tex]:
Since [tex]\(c\)[/tex] is already under the square root:
[tex]\[
\sqrt{c}
\][/tex]
- For [tex]\(\sqrt{d^8}\)[/tex]:
[tex]\[
\sqrt{d^8} = \sqrt{(d^4)^2}
\][/tex]
Since [tex]\(d^8\)[/tex] is a perfect square ([tex]\((d^4)^2\)[/tex]):
[tex]\[
\sqrt{d^8} = d^4
\][/tex]
3. Combine all the simplified square roots:
Now, we combine all the simplified terms:
[tex]\[
\sqrt{a^5 b^3 c d^8} = (a^2 \sqrt{a})(b \sqrt{b})(\sqrt{c})(d^4)
\][/tex]
4. Combine the radical parts and the non-radical parts:
- The non-radical parts are [tex]\(a^2\)[/tex], [tex]\(b\)[/tex], and [tex]\(d^4\)[/tex].
- The radical parts are [tex]\(\sqrt{a}\)[/tex], [tex]\(\sqrt{b}\)[/tex], and [tex]\(\sqrt{c}\)[/tex].
So, we write:
[tex]\[
\sqrt{a^5 b^3 c d^8} = a^2 b d^4 \cdot \sqrt{a b c}
\][/tex]
Therefore, the simplified form of the expression [tex]\(\sqrt{a^5 b^3 c d^8}\)[/tex] is:
[tex]\[
a^2 b d^4 \sqrt{a b c}
\][/tex]
