Question

How do you solve ${4^{x - 3}} = \dfrac{1}{{16}}$?

Answer 1

Hint: In this question, we need to find the solution for the exponential equation. Here we try to make the bases same on both sides of the equation by some basic calculations. Firstly, we replace 16 by square of 4, i.e. ${4^2}$. And then we use the negative exponent rule given by $\dfrac{1}{{{a^n}}} = {a^{ - n}}$. After making this, we get the same base on both sides which is 4. Then we use the fact that, if the bases are the same then the equation is equal only if the exponents are also equal and find the required solution.

Complete step by step solution:
Given an equation of the form,
${4^{x - 3}} = \dfrac{1}{{16}}$ …… (1)
We are asked to solve the exponential equation given in the equation (1).
Note that we approach the above exponential equation, by making the bases same on both sides.
We know that 16 is a perfect square and it is a power of 4.
i.e. $16 = {4^2}$.
So we replace 16 in the denominator on the R.H.S. we get,
$ \Rightarrow {4^{x - 3}} = \dfrac{1}{{{4^2}}}$ …… (2)
Now we move ${4^2}$ to the numerator using the negative exponent rule which is given by,
$\dfrac{1}{{{a^n}}} = {a^{ - n}}$
Here $a = 4$ and $n = 2$
Hence the equation (2) becomes,
$ \Rightarrow {4^{x - 3}} = {4^{ - 2}}$
Now the base on both sides of the equation is the same which is 4.
We have the result that if the expression having the same bases is equal only if the exponents or indices are also equal.
Hence equating the indices, we get,
$ \Rightarrow x - 3 = - 2$
Move all the terms which do not contain the x term to the R.H.S.
Transferring -3 to the other side we get,
$ \Rightarrow x = - 2 + 3$
$ \Rightarrow x = 1$

Hence the solution for the equation ${4^{x - 3}} = \dfrac{1}{{16}}$ is $x = 1$.

Note: We can verify the obtained answer is correct by substituting back it in the given equation. If the equation satisfies, i.e. if we get L.H.S. is equal to R.H.S. then the obtained value is correct. Otherwise our answer is wrong.
Students must remember the rules of exponents to simplify such problems. We need to be careful while applying the rules. It is necessary to use the correct rule to split the terms and simplify the answer.
The rules of exponents are given below.
(1) Multiplication rule : ${a^m} \cdot {a^n} = {a^{m + n}}$
(2) Division rule : $\dfrac{{{a^m}}}{{{a^n}}} = {a^{m - n}}$
(3) Power of a power rule : ${({a^m})^n} = {a^{mn}}$
(4) Power of a product rule : ${(ab)^m} = {a^m}{b^m}$
(5) Power of a fraction rule : ${\left( {\dfrac{a}{b}} \right)^m} = \dfrac{{{a^m}}}{{{b^m}}}$
(6) Zero exponent : ${a^0} = 1$
(7) Negative exponent : ${a^{ - x}} = \dfrac{1}{{{a^x}}}$