Question

Choose the correct alternative in the following question:

$$\begin{gathered} {\left(8^4+8^2\right)}^{\frac{1}{2}}= \\ \left(\mathrm{a}\right)84\left(\mathrm{b}\right)8\sqrt{77}\left(\mathrm{c}\right)72\left(\mathrm{d}\right)8\sqrt{65} \end{gathered}$$

Answer 1

Since,

$$\begin{gathered} {\left(8^4+8^2\right)}^{\frac{1}{2}} \\ ={\left(8^{2+2}+8^2\right)}^{\frac{1}{2}} \\ ={\left(8^2\times 8^2+8^2\right)}^{\frac{1}{2}}\left(\mathrm{As},x^{m+n}=x^m\times x^n\right) \\ ={\left[8^2\times \left(8^2+1\right)\right]}^{\frac{1}{2}}\left[\mathrm{As},ab+ac=a\times \left(b+c\right)\right] \\ ={\left(8^2\right)}^{\frac{1}{2}}\times {\left(8^2+1\right)}^{\frac{1}{2}}\left[\mathrm{As},{\left(ab\right)}^m=a^m\times b^m\right] \\ =8^{2\times \frac{1}{2}}\times {\left(64+1\right)}^{\frac{1}{2}} \\ =8\times {65}^{\frac{1}{2}} \\ =8\sqrt{65} \end{gathered}$$

Hence, the correct alternative is option (d).