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Simplify $ \sqrt{a^{3} b^{4}} \cdot \sqrt[3]{a^{4} b^{3}} $.
Given:
\( \sqrt{a^{3} b^{4}} \cdot \sqrt[3]{a^{4} b^{3}} \).
To do:
We have to simplify \( \sqrt{a^{3} b^{4}} \cdot \sqrt[3]{a^{4} b^{3}} \).
Solution:
We know that,
$a^m \times a^n=a^{m+n}$
Therefore,
$\sqrt{a^{3} b^{4}} \cdot \sqrt[3]{a^{4} b^{3}}=a^{\frac{3}{2}}b^{\frac{4}{2}} \cdot a^{\frac{4}{3}}b^{\frac{3}{3}}$
$=a^{\frac{3}{2}}b^1 \cdot a^{\frac{4}{3}}b^1$
$=a^{\frac{3}{2}+\frac{4}{3}} \cdot b^{1+1}$
$=a^{\frac{3\times3+2\times4}{6}}b^2$
$=a^{\frac{17}{6}}b^2$
Hence, $\sqrt{a^{3} b^{4}} \cdot \sqrt[3]{a^{4} b^{3}}=a^{\frac{17}{6}}b^2$.
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