Chapter 6.5, Problem 10E

To determine

To solve the equation:

  ${216}^x=6^{x+10}$

And check the solution.

Answer to Problem 10E

  $x=5$

Explanation of Solution

Given:

  ${216}^x=6^{x+10}$

Concept Used:

Two powers with the same positive base(not equals to 1), are equal if and only if their exponents are equal, that is:

  $b^x=b^y\iff x=y;\text{ where }b>0,\text{ and }b\ne 1$

Calculation:

To solve the equation:

  ${216}^x=6^{x+10}$

First, simplify the equation so the base of the equation is same, as

  $\begin{array}{c}{216}^x=6^{x+10}\\ \Rightarrow {\left(6^3\right)}^x=6^{x+10};\because 6^3=216\\ \Rightarrow 6^{3x}=6^{x+10}\end{array}$

Now, since base of above powers are same; thus

  $\begin{array}{c}{6}^{3x}=6^{x+10}\\ \Rightarrow 3x=x+10\\ \Rightarrow 3x-x=10\\ \Rightarrow 2x=10\\ \Rightarrow x=\frac{10}{2}\\ \Rightarrow x=5\end{array}$

Thus, the solution of given equation is:

  $x=5$

Now, to check this solution substitute $x=5$ in given equation and check whether both sides are equal or not, as

  $\begin{array}{c}{216}^5=6^{5+10}\\ {216}^5=6^{15}\\ {216}^5=6^{3\times 5}\\ {216}^5={\left(6^3\right)}^5\\ \Rightarrow {216}^5={216}^5;\because 6^3=216\end{array}$

Since, both sides of above equation becomes equal, thus $x=5$ satisfies the given equation.