To simplify the expression [tex]\(\sqrt{50 q^9}\)[/tex], we follow these steps:
1. Factor the expression under the square root:
[tex]\[
\sqrt{50 q^9}
\][/tex]
We recognize that 50 can be factored into prime factors as [tex]\(50 = 25 \cdot 2\)[/tex], and [tex]\(25\)[/tex] is a perfect square. So we rewrite the expression as:
[tex]\[
\sqrt{25 \cdot 2 \cdot q^9}
\][/tex]
2. Split the square root into separate square roots for each factor:
[tex]\[
\sqrt{25 \cdot 2 \cdot q^9} = \sqrt{25} \cdot \sqrt{2} \cdot \sqrt{q^9}
\][/tex]
3. Simplify the square root of the perfect square 25:
[tex]\[
\sqrt{25} = 5
\][/tex]
So our expression becomes:
[tex]\[
5 \cdot \sqrt{2} \cdot \sqrt{q^9}
\][/tex]
4. Simplify the square root of [tex]\(q^9\)[/tex]:
[tex]\[
\sqrt{q^9} = q^{9/2}
\][/tex]
since the square root of [tex]\(q^n\)[/tex] is [tex]\(q^{n/2}\)[/tex].
5. Combine all parts together:
[tex]\[
5 \cdot \sqrt{2} \cdot q^{9/2}
\][/tex]
Thus, the simplified form of the expression [tex]\(\sqrt{50 q^9}\)[/tex] is:
[tex]\[
5\sqrt{2} \sqrt{q^9}
\][/tex]
And since [tex]\(\sqrt{q^9} = q^{9/2}\)[/tex], we can express the final answer as:
[tex]\[
5\sqrt{2} \, q^{9/2}
\][/tex]
In this specific case, keeping [tex]\(5\sqrt{2}\sqrt{q^9}\)[/tex] is already simplified and sufficient.
