Question

How do you express $\dfrac{81}{9}$ as a power of $3$?

Answer 1

Hint: : In the above question, we are given a fraction $\dfrac{81}{9}$ to be written as a power of $3$. For this, we have to use the properties of exponents so as to write both of the numerator and the denominator as the powers of $3$. Both the numerator and the denominator can be written as the powers of $3$. The numerator $81$ can be written as the fourth power of $3$, or ${{3}^{4}}$ and the denominator $9$ can be written as the square of $3$, or ${{3}^{2}}$. So the given fraction will become $\dfrac{{{3}^{4}}}{{{3}^{2}}}$. Then by using the exponent property given by $\dfrac{{{a}^{m}}}{{{a}^{n}}}={{a}^{m-n}}$, we can simplify the fraction $\dfrac{{{3}^{4}}}{{{3}^{2}}}$ and write it as a power of $3$.

Complete step-by-step solution:
Let us write the given fraction as
$\Rightarrow x=\dfrac{81}{9}$
According to the above question, we have to write the given fraction as a power of $3$. For this, we write both the numerator and the denominator as the powers of $3$. We know that the numerator $81$ is equal to the fourth power of three. Also, the denominator $9$ can be written as the square of three. So we can put $81={{3}^{4}}$ and $9={{3}^{2}}$ in the above fraction to get
$\Rightarrow x=\dfrac{{{3}^{4}}}{{{3}^{2}}}$
Now, from the properties of the exponents we know that $\dfrac{{{a}^{m}}}{{{a}^{n}}}={{a}^{m-n}}$. Therefore, the above fraction can also be written as
\[\begin{align}
  & \Rightarrow x={{3}^{4-2}} \\
 & \Rightarrow x={{3}^{2}} \\
\end{align}\]
Hence, the given fraction of $\dfrac{81}{9}$ can be written as a power of $3$ as ${{3}^{2}}$.

Note: For solving these types of questions, we must be familiar with the properties of the exponents. We can also solve this question by considering the division of $81$ by $9$, which will give us $9$. And since $9$ is a square of $3$, so the given fraction will become ${{3}^{2}}$.