Question

Simplify \[{9^{\dfrac{3}{2}}}\].

Answer 1

Hint:
Here we will first split the exponent of the given exponent. Then we will simplify and solve the expression with the help of the basic arithmetic operations i.e. square root of a number and cube of a number. Then we will get the value of the given expression.

Complete Step by step Solution:
Given expression is \[{9^{\dfrac{3}{2}}}\].
Firstly we will split the exponent of the given expression into two numbers. Therefore, we get
We can write \[\dfrac{3}{2}\] as \[\dfrac{1}{2} \times 3\] which is the same thing.
So using this and modifying the given equation, we get
\[ \Rightarrow {9^{\dfrac{3}{2}}} = {9^{\dfrac{1}{2} \times 3}}\]
We can also write it as
\[ \Rightarrow {9^{\dfrac{3}{2}}} = {\left( {{9^{\dfrac{1}{2}}}} \right)^3}\]
Now we will solve this equation to get its value. We know that when the exponent of a number is \[\dfrac{1}{2}\], which means that it’s the square root of the number. Therefore, we get
\[ \Rightarrow {9^{\dfrac{3}{2}}} = {\left( {\sqrt 9 } \right)^3}\]
Now we know that the square root of the number 9 is equal to \[ \pm 3\] i.e. \[\sqrt 9 = \pm 3\]. Therefore, we get
\[ \Rightarrow {9^{\dfrac{3}{2}}} = {\left( { \pm 3} \right)^3}\]
Now by solving the above equation, we get
\[ \Rightarrow {9^{\dfrac{3}{2}}} = {\left( { \pm 3} \right)^3} = \pm 27\]

Hence the value of the given expression \[{9^{\dfrac{3}{2}}}\] is equal to \[ \pm 27\].

Note:
Exponent is the defined as the number which represents how many times a number is being multiplied to itself. If the exponent of a number is zero then the value of the number is 1 i.e. \[{a^0} = 1\]. Cube root of a number is the factor that we multiply by itself three times to get that number. We don’t have to confuse the cube root with the square root. The square root of a number is the factor that we multiply by itself two times to get that number.
Square root is expressed as \[\sqrt[2]{{{\rm{number}}}}\]
Cube root is expressed as \[\sqrt[3]{{{\rm{number}}}}\]