Q. 158 - Intermediate Algebra

Question

In the following exercises,simplify
$$\begin{gathered} \left(a\right) {(625 n^{\frac{8}{3}})}^{\frac{3}{4}} \\ \left(b\right) {(9 x^{\frac{2}{5}} y^{\frac{3}{5}})}^{\frac{5}{2}} \end{gathered}$$

Step-by-Step Solution

Answer

$$\begin{gathered} \left(a\right) 125 u^2 \\ \left(b\right) 243 x{y}^{\frac{3}{2}} \end{gathered}$$

Part (a) Step 1.Given information

Consider the given expression${(625 n^{\frac{8}{3}})}^{\frac{3}{4}}$

To simplify the expression,by used power property that is ${\left(a^m\right)}^n=a^{m.n}$.

Part (a) Step 2. Simplify the expression

$$\begin{gathered} {(625 n^{\frac{8}{3}})}^{\frac{3}{4}}={\left(625\right)}^{\frac{3}{4}}.{\left(n^{\frac{8}{3}}\right)}^{\frac{3}{4} [\sin ce ;{\left(ab\right)}^m={\left(a\right)}^m.{\left(b\right)}^m]} \\ ={\left(5^4\right)}^{\frac{3}{4}}.{\left(n^{\frac{8}{3}}\right)}^{\frac{3}{4} [\sin ce; 5^4=625]} \\ ={\left(5\right)}^{4.\frac{3}{4}}{\left(n\right)}^{\frac{3}{4}\frac{8}{3} [\sin ce;{\left(a^m\right)}^n=a^{mn}]} \\ ={\left(5\right)}^{\frac{12}{4}}.{\left(u\right)}^{\frac{24}{12}} [Multiplay the exponent] \\ ={\left(5\right)}^3.{\left(u\right)}^{2 } [Divide the exponent] \\ =125 u^{2 } \left[Simplify\right] \\ Therefore,the answer is 125 u^2 \end{gathered}$$

Part (b) Step 1.Given information

Consider the given expression ${(9 x^{\frac{2}{5}} y^{\frac{3}{5}})}^{\frac{5}{2}}$

To simplify the expression,by used power property that is ${\left(a^m\right)}^n=\left(a^{mn}\right)$.

Part (b) Step 2.Simplify the expression

$$\begin{gathered} {(9 x^{\frac{2}{5}} y^{\frac{3}{5}})}^{\frac{5}{2}}=\left(9^{\frac{5}{2}}\right).{\left(x^{\frac{2}{5}}\right)}^{\frac{5}{2}}.{\left(y^{\frac{3}{5}}\right)}^{\frac{5}{2}} [Since;{\left(ab\right)}^m={\left(a\right)}^m.{\left(b\right)}^{m]} \\ ={\left(3^2\right)}^{\frac{5}{2}}.{\left(x^{\frac{2}{5}}\right)}^{\frac{5}{2}}.{\left(y^{\frac{3}{5}}\right)}^{\frac{5}{2} [Since;2^2=4]} \\ ={\left(3\right)}^{2.\frac{5}{2}}{\left(x\right)}^{\frac{5}{2}.\frac{2}{5}}.{\left(y\right)}^{\frac{5}{2}.\frac{3}{5} [Since {\left(a^m\right)}^n=a^{m.n}]} \\ ={\left(3\right)}^{\frac{10}{2}}.{\left(x\right)}^{\frac{10}{10}}.{\left(y\right)}^{\frac{15}{10}} [Multiplay the exponent] \\ =3^5.x.y^{\frac{3}{2}} [Divide the exponent] \\ =243x{y}^{\frac{3}{2}} [simplify] \\ Therefore,the answer is 243x{y}^{\frac{3}{2}}. \end{gathered}$$