Question

Find the square root of 81.
A. 9
B. 45
C. 18
D. 81

Answer 1

Hint: We try to form the number in the form of the power of a digit. We convert 81 to the power value of prime 3. Then we perform the theorem of indices of ${{\left( {{a}^{x}} \right)}^{b}}={{a}^{bx}}$ to find the root value using algebra. We also can use an elementary method to find the root taking only 1 prime out of 2 similar primes.

Complete step-by-step solution
We need to find the root of 81.
If the root of the digit y is x then we can say ${{x}^{2}}=y$ or $x=\sqrt{y}$.
We need to find the prime factorization of 81 to find its root.
So, $81={{3}^{4}}$. To find the root value we take 1 similar prime value out of 2.
$\sqrt{81}=\sqrt{3\times 3\times 3\times 3}=3\times 3=9$.
We know that ${{\left( {{a}^{x}} \right)}^{b}}={{a}^{bx}}$.
We also can use indices where $\sqrt{81}=\sqrt{{{3}^{4}}}={{\left( {{3}^{4}} \right)}^{\dfrac{1}{2}}}={{3}^{4\times \dfrac{1}{2}}}={{3}^{2}}=9$.
The correct option is A.

Note: If we take the root value the negative part of the value also can be the answer. Now if we form the polynomial equation then we get ${{x}^{2}}=y=81$. We solve it using the formula ${{a}^{2}}-{{b}^{2}}=\left( a+b \right)\left( a-b \right)$.
$\begin{align}
  & {{x}^{2}}=81 \\
 & \Rightarrow {{x}^{2}}-81=0 \\
 & \Rightarrow \left( x+9 \right)\left( x-9 \right)=0 \\
\end{align}$
Solving the equation, we get two values of x as $x=9,-9$. Both are roots of 81.