Let's solve the expression [tex]\(\sqrt[4]{9 m^5} \div \sqrt[3]{3 m^2}\)[/tex] step by step.
### Step 1: Simplify each term
#### Term 1: [tex]\(\sqrt[4]{9 m^5}\)[/tex]
- This expression can be rewritten using exponent notation. Recall that the [tex]\(n\)[/tex]-th root of a number [tex]\(a\)[/tex] is [tex]\(a^{1/n}\)[/tex].
- Therefore, [tex]\(\sqrt[4]{9 m^5} = (9 m^5)^{1/4}\)[/tex].
We can separate this into two parts:
[tex]\[ 9^{1/4} \times (m^5)^{1/4}. \][/tex]
- Simplify each part:
- [tex]\(9^{1/4}\)[/tex]: Since [tex]\(9 = 3^2\)[/tex], we have [tex]\(9^{1/4} = (3^2)^{1/4} = 3^{2 \times (1/4)} = 3^{1/2} = \sqrt{3}\)[/tex].
- [tex]\((m^5)^{1/4} = m^{5 \times (1/4)} = m^{5/4}\)[/tex].
So, [tex]\(\sqrt[4]{9 m^5} = \sqrt{3} \cdot m^{5/4}\)[/tex].
#### Term 2: [tex]\(\sqrt[3]{3 m^2}\)[/tex]
- Similarly, rewrite the expression as [tex]\((3 m^2)^{1/3}\)[/tex].
Separate it into two parts:
[tex]\[ 3^{1/3} \times (m^2)^{1/3}. \][/tex]
- Simplify each part:
- [tex]\(3^{1/3}\)[/tex]: This is simply the cube root of 3.
- [tex]\((m^2)^{1/3} = m^{2 \times (1/3)} = m^{2/3}\)[/tex].
So, [tex]\(\sqrt[3]{3 m^2} = 3^{1/3} \cdot m^{2/3}\)[/tex].
### Step 2: Divide the simplified terms
Now we need to divide [tex]\(\sqrt{3} \cdot m^{5/4}\)[/tex] by [tex]\(3^{1/3} \cdot m^{2/3}\)[/tex].
[tex]\[
\frac{\sqrt{3} \cdot m^{5/4}}{3^{1/3} \cdot m^{2/3}}.
\][/tex]
- Simplify the numerical part:
[tex]\[
\frac{\sqrt{3}}{3^{1/3}} = 3^{1/2 - 1/3} = 3^{(1/2) - (1/3)} = 3^{3/6 - 2/6} = 3^{1/6}.
\][/tex]
- Simplify the [tex]\(m\)[/tex]-term:
[tex]\[
\frac{m^{5/4}}{m^{2/3}} = m^{5/4 - 2/3} = m^{5/4 - 2/3} = m^{15/12 - 8/12} = m^{7/12}.
\][/tex]
Putting it all together, we get:
[tex]\[
\frac{\sqrt{3} \cdot m^{5/4}}{3^{1/3} \cdot m^{2/3}} = 3^{1/6} \cdot m^{7/12}.
\][/tex]
### Final Answer
Thus, the simplified form of the expression [tex]\(\sqrt[4]{9 m^5} \div \sqrt[3]{3 m^2}\)[/tex] is:
[tex]\[
3^{1/6} \cdot m^{7/12}.
\][/tex]
