Question

How do you solve the expression ${{8}^{x-3}}={{16}^{x}}$ ?

Answer 1

Hint: We need to find the value of $x$ in the given equation. We start to solve the given question by expressing the given terms to the power of 2. Then, we equate the powers of the terms with the same base according to the rules of exponents. Lastly, we simplify the expression to get the desired result.

Complete step-by-step solution:
We are given an equation and need to find the value of $x$ in it. We will be solving the given question using the rules of exponents.
Exponents, in mathematics, refer to the number of times a number is multiplied by itself. It is written above the base number to the right.
The general representation of the exponents is given by
$\Rightarrow {{a}^{x}}=a\times a\times a.....\times a$
Here,
$a=$ base
$x=$ exponent
In exponential equations, the powers of the terms must be equal if the bases of the terms are equal.
Writing the above lines in the form of an equation, we get,
$\Rightarrow {{a}^{n}}={{a}^{m}}$
From the above, we conclude that $n=m$
Let us understand the working of the above rule with an example,
Example:
Find the value of ${{2}^{2x}}={{2}^{6}}?$
In the given question, the bases of both terms are the same.
The result is obtained just by equating the corresponding exponents of the terms.
Applying the same, we get,
$\Rightarrow 2x=6$
$\Rightarrow x=\dfrac{6}{2}$
$\therefore x=3$
According to our question,
$\Rightarrow {{8}^{x-3}}={{16}^{x}}$
We need to make the bases of the terms equal. Each of the terms given in the equation can be expressed to the power of 2.
Expressing the terms to the power of 2, we get,
$\Rightarrow {{\left( {{2}^{3}} \right)}^{x-3}}={{\left( {{2}^{4}} \right)}^{x}}$
From the rules of exponents, we know that ${{\left( {{a}^{m}} \right)}^{n}}={{\left( a \right)}^{mn}}$
Applying the same, we get,
$\Rightarrow {{2}^{3x-9}}={{2}^{4x}}$
For the above equation, the bases of the terms are the same.
From the rules of exponential equations, we know that when bases are the same, powers can be equated.
Following the same, we get,
$\Rightarrow 3x-9=4x$
Shifting all the terms containing x to one side of the equation, we get,
$\Rightarrow 3x-4x=9$
Simplifying the above equation, we get,
$\Rightarrow -x=9$
Multiplying with the minus sign on both sides of the equation, we get,
$\Rightarrow -\left( -x \right)=-9$
From rules of arithmetic, we know that $\left( - \right)\times \left( - \right)=\left( + \right)$
Following the same, we get,
$\therefore x=-9$

Note: The result of the given question can be cross-checked by substituting the value of x in the given equation.
LHS:
$\Rightarrow {{8}^{x-3}}$
Substituting the value of x, we get,
$\Rightarrow {{8}^{-9-3}}$
$\Rightarrow {{8}^{-12}}$
Expressing to the power of 2, we get,
$\Rightarrow {{\left( {{2}^{3}} \right)}^{-12}}$
Multiplying the powers, we get,
$\Rightarrow {{2}^{-36}}$
RHS:
$\Rightarrow {{16}^{x}}$
Substituting the value of x, we get,
$\Rightarrow {{16}^{-9}}$
Expressing to the power of 2, we get,
$\Rightarrow {{\left( {{2}^{4}} \right)}^{-9}}$
Multiplying the powers, we get,
$\Rightarrow {{2}^{-36}}$
LHS $=\;$ RHS
The result attained is correct.