To simplify the given expression [tex]\(\left(49 x^2 y\right)^{\frac{1}{2}}\left(27 x^6 y^{4}\right)^{\frac{1}{3}}\)[/tex]:
1. Simplify each term separately:
- For [tex]\((49 x^2 y)^{\frac{1}{2}}\)[/tex]:
[tex]\[
(49 x^2 y)^{\frac{1}{2}} = \sqrt{49 x^2 y}
\][/tex]
Since [tex]\(\sqrt{49} = 7\)[/tex] and [tex]\(\sqrt{x^2} = x\)[/tex]:
[tex]\[
\sqrt{49 x^2 y} = 7 \sqrt{x^2 y} = 7 \sqrt{x^2} \sqrt{y} = 7 x \sqrt{y} \text{ (because \(x\) is assumed non-negative hence }\sqrt{x^2} = x)
\][/tex]
- For [tex]\((27 x^6 y^4)^{\frac{1}{3}}\)[/tex]:
[tex]\[
(27 x^6 y^4)^{\frac{1}{3}} = 27^{\frac{1}{3}} (x^6)^{\frac{1}{3}} (y^4)^{\frac{1}{3}}
\][/tex]
Since [tex]\(27^{\frac{1}{3}} = 3\)[/tex], [tex]\((x^6)^{\frac{1}{3}} = x^2\)[/tex], and [tex]\((y^4)^{\frac{1}{3}} = y^{\frac{4}{3}}\)[/tex]:
[tex]\[
(27 x^6 y^4)^{\frac{1}{3}} = 3 x^2 y^{\frac{4}{3}}
\][/tex]
2. Multiply the simplified terms together:
[tex]\[
7 x \sqrt{y} \cdot 3 x^2 y^{\frac{4}{3}}
\][/tex]
3. Combine the coefficients and like terms:
[tex]\[
7 \cdot 3 \cdot x \cdot x^2 \cdot \sqrt{y} \cdot y^{\frac{4}{3}}
\][/tex]
- Combine the constants:
[tex]\[
7 \cdot 3 = 21
\][/tex]
- Combine the [tex]\(x\)[/tex] terms:
[tex]\[
x \cdot x^2 = x^3
\][/tex]
- Combine the [tex]\(y\)[/tex] terms. Note that [tex]\(\sqrt{y}\)[/tex] is [tex]\(y^{1/2}\)[/tex], so:
[tex]\[
y^{\frac{1}{2}} \cdot y^{\frac{4}{3}} = y^{\frac{1}{2} + \frac{4}{3}}
\][/tex]
Find a common denominator for the exponents:
[tex]\[
\frac{1}{2} + \frac{4}{3} = \frac{3}{6} + \frac{8}{6} = \frac{11}{6}
\][/tex]
Thus, the final simplified expression is:
[tex]\[
21 x^3 y^{\frac{11}{6}}
\][/tex]
Therefore, [tex]\(\left(49 x^2 y\right)^{\frac{1}{2}}\left(27 x^6 y^{\frac{8}{2}}\right)^{\frac{1}{3}} = 21 x^3 y^{\frac{11}{6}}\)[/tex].
