Question

Find the value of: ${9^{\dfrac{3}{2}}}$

Answer 1

Hint: We can see this problem is from indices and powers. This number given is having $9$ as base and $\left( {\dfrac{3}{2}} \right)$ as power. But since we have to simplify this we can write $\left( {\dfrac{3}{2}} \right)$ as the cube of the square-root of the given number in base. So, we first have to find out the square root of the base number, $9$ and then the cube of the square root of $4$. The result thus obtained will be the answer to the given question.

Complete step by step solution:
So, the given question requires us to simplify $9$ to the power $\left( {\dfrac{3}{2}} \right)$. $9$ to the power $\left( {\dfrac{3}{2}} \right)$ can be written as ${9^{\dfrac{3}{2}}}$ .
This is of the form ${a^{\left( {\dfrac{m}{n}} \right)}}$. But we can rewrite the expression using the laws of indices and powers as ${a^{\left( {\dfrac{m}{n}} \right)}} = {\left( {{a^m}} \right)^{\dfrac{1}{n}}}$ .
Thus we will apply same on the question above, we get,
${9^{\dfrac{3}{2}}} = {\left( {{9^{\dfrac{1}{2}}}} \right)^3}$
Now we first solve the power inside the bracket. Now, the power $\left( {\dfrac{1}{2}} \right)$ represents the square-root of the entity. To find the square root of $9$, we have to first factorise the number. We know that $9 = 3 \times 3 = \left( {{3^2}} \right)$
$ = {\left( {{{\left( {{3^2}} \right)}^{\dfrac{1}{2}}}} \right)^3}$
Now, This is of the form ${\left( {{a^m}} \right)^{\dfrac{1}{n}}}$. But we can rewrite the expression using the laws of indices and powers ${a^{\left( {\dfrac{m}{n}} \right)}} = {\left( {{a^m}} \right)^{\dfrac{1}{n}}}$ . So, we have,
$ = {\left( {{3^{2 \times \dfrac{1}{2}}}} \right)^3}$
Simplifying the calculations, we get,
$ = {\left( 3 \right)^3}$
Now, we have to find the cube of the number $3$.So, we get,
$ = \left( 3 \right) \times \left( 3 \right) \times \left( 3 \right)$
$ = 27$
So, the correct answer is “27”.

Note: These rules or laws of indices help us to minimize the problems and get the answer in very less time. These powers can be positive and negative but can be moulded according to our convenience while solving the problem. Also note that cube-root, square-root are fractions with 1 as numerator and respective root in denominator.