Question

Fill in the blank to make the statement true:

${\left(\frac{11}{15}\right)}^4\times {\left(\_\_\_\right)}^5={\left(\frac{11}{15}\right)}^9$

Answer 1

Find the missing number.

Given equation: ${\left(\frac{11}{15}\right)}^4\times {\left(\_\_\_\right)}^5={\left(\frac{11}{15}\right)}^9$

We know that for any non-zero integers $‘a’$whole numbers $m$ and $n$, $a^m\times a^n=a^{m+n}$ and $a^m÷a^n=a^{m-n}$

Let the missing number be $x$.

So the given equation becomes,

$$\begin{gathered} {\left(\frac{11}{15}\right)}^4\times {\left(x\right)}^5={\left(\frac{11}{15}\right)}^9 \\ \Rightarrow {\left(x\right)}^5=\frac{{\left(\frac{11}{15}\right)}^9}{{\left(\frac{11}{15}\right)}^4} \\ \Rightarrow {\left(x\right)}^5={\left(\frac{11}{15}\right)}^{9-4}\mathbf{}\mathbf{[}\mathbf{\because }{\mathit{a}}^{\mathbf{m}}\mathbf{}\mathbf{÷}\mathbf{}{\mathit{a}}^{\mathbf{n}}\mathbf{}\mathbf{=}\mathbf{}{\mathit{a}}^{\mathbf{m}\mathbf{-}\mathbf{n}}\mathbf{]} \\ \Rightarrow {\left(x\right)}^5={\left(\frac{11}{15}\right)}^5 \\ \Rightarrow x=\frac{11}{15} \end{gathered}$$

Hence, ${\left(\frac{11}{15}\right)}^4\times {\left(\underline{\frac{11}{15}}\right)}^5={\left(\frac{11}{15}\right)}^9$