Question

How do you solve ${{2}^{x+6}}=32$ ?

Answer 1

Hint: Here, we have to find the value of ‘x’ in the above expression. As we already know about the properties of exponents and powers. Here we are using the property of equating the powers of both the sides. For that, first we need to find the LCM of 32. When we will get the LCM, we will express 32 in exponential form. Exponential form means to express the term in the form of exponent and power. Just like this: ${{2}^{12}},{{5}^{44}}$ etc.

Complete step-by-step solution:
Now let’s come to the question and see how to solve it.
First, write the expression given in question:
$\Rightarrow {{2}^{x+6}}=32$
Now, we need to find the LCM of 32 because we have to make the base equal of both sides in order to equate the powers.
$\Rightarrow $ LCM of 32:
$\begin{align}
  & 2\left| \!{\underline {\,
  32 \,}} \right. \\
 & 2\left| \!{\underline {\,
  16 \,}} \right. \\
 & 2\left| \!{\underline {\,
  8 \,}} \right. \\
 & 2\left| \!{\underline {\,
  4 \,}} \right. \\
 & 2\left| \!{\underline {\,
  2 \,}} \right. \\
 & 1\left| \!{\underline {\,
  1 \,}} \right. \\
\end{align}$
$\Rightarrow 2\times 2\times 2\times 2\times 2$
$\Rightarrow {{2}^{5}}$
Here, ${{2}^{5}}$is the LCM of 32 in exponential form.
So, let’s place ${{2}^{5}}$ in the given expression.
Now, our expression will be:
$\Rightarrow {{2}^{x+6}}={{2}^{5}}$
Now, have a look at the expression and observe that we got the base equal of both the sides in order to equate the powers.
Finally, we are equating the powers of both the sides so that we can find the value of ‘x’ in the expression.
$\Rightarrow $ x + 6 = 5
$\Rightarrow $ x = 5 – 6
$\therefore $ x = -1
So, we have got the value of ‘x’ as -1.

Note: There is an alternative method in order to solve this question which is quite similar. For that method one should know the property ${{a}^{m}}\div {{a}^{n}}={{a}^{m-n}}$ and ${{a}^{m}}\times {{a}^{n}}={{a}^{m+n}}$
So all the steps are following:
$\Rightarrow {{2}^{x}}\times {{2}^{6}}=32$ By using ${{a}^{m}}\times {{a}^{n}}={{a}^{m+n}}$
$\Rightarrow $ By taking LCM of 32 as shown in above method, expression will be:
$\Rightarrow {{2}^{x}}\times {{2}^{6}}={{2}^{5}}$
$\Rightarrow {{2}^{x}}=\dfrac{{{2}^{5}}}{{{2}^{6}}}$
Now by using ${{a}^{m}}\div {{a}^{n}}={{a}^{m-n}}$:
$\Rightarrow {{2}^{x}}={{2}^{5-6}}$
On further solving,
$\Rightarrow {{2}^{x}}={{2}^{-1}}$
Now, the value of ‘x’ will be -1 as the base on both sides is the same.
Apart from this, taking LCM is most important because without that step, we can’t equate the powers and find the value of ‘x’. The easiest method is the first one to solve the question.