Question

Solve the algebraic expression ${{7}^{x}}=\dfrac{1}{49}$?

Answer 1

Hint: To solve the given equation we have to simplify the given equation furthermore. Certain transformations and substitutions should be made to get the equation simplified and to get the value of $x$. And finally verification should be done to check whether the value of $x$ we get is correct or not.

Complete step-by-step solution:
From the question it had been given that, ${{7}^{x}}=\dfrac{1}{49}$
Now, as we have been discussed already above we have to do some transformations and substitutions to get the equation into simplified form.
${{7}^{x}}=\dfrac{1}{49}$
${{7}^{x}}=\dfrac{1}{{{7}^{2}}}$
${{7}^{x}}\times {{7}^{2}}=1$
Now, we know that zero power of any number is $1$.
Therefore, $1$ can be written as ${{7}^{0}}$
${{7}^{x}}\times {{7}^{2}}={{7}^{0}}$
Now, we also know some formulae for exponents with the same bases.
${{a}^{m}}\times {{a}^{n}}={{a}^{m+n}}$
By using the above formula we can further simplify the equation.
${{7}^{x+2}}={{7}^{0}}$
Now, as the bases are equal we have to equate the powers, then we can get the value of $x$
$\begin{align}
  & x+2=0 \\
 & x=-2 \\
\end{align}$
Therefore, the value of $x=-2$
Now we have to verify whether we got the exact value or not.
Verification:
First we have to verify the left hand side of the equation.
Left hand side:
${{7}^{x}}={{7}^{-2}}$
${{7}^{x}}=\dfrac{1}{49}$
We know that right hand side of the equation is $\dfrac{1}{49}$
We can clearly observe that the left hand side of the equation is equal to the right hand side of the equation.
Hence, verified.

Note: While answering questions of this type, there are some general rules of exponents that every student must know. They are ${{a}^{m+n}}={{a}^{m}}\times {{a}^{n}}$, ${{a}^{m-n}}=\dfrac{{{a}^{m}}}{{{a}^{n}}}$, ${{\left( {{a}^{x}} \right)}^{y}}={{a}^{xy}}$ , $\dfrac{{{a}^{x}}}{{{b}^{x}}}={{\left( \dfrac{a}{b} \right)}^{x}}$ , ${{a}^{0}}=1$ , ${{a}^{-x}}=\dfrac{1}{{{a}^{x}}}$ and ${{a}^{\dfrac{x}{y}}}=\sqrt[y]{{{a}^{x}}}$ .These rules we should remember which will help us to solve them efficiently.