Question

How do you simplify \[{\left( {\dfrac{1}{2}} \right)^{\dfrac{5}{2}}}\]?

Answer 1

Hint: Here, we will first rewrite the exponent which is \[\dfrac{5}{2}\] as a product of \[5\] and \[\dfrac{1}{2}\]. Then, we will raise the base \[\dfrac{1}{2}\] to the exponent \[\dfrac{1}{2}\]. Finally, we will raise the resultant value to the exponent \[5\] to get the required answer.

Formula used:
Law of exponents: \[{\left( {{a^m}} \right)^n} = {a^{mn}}\]

Complete step by step solution:
We are required to simplify and find the value of \[{\left( {\dfrac{1}{2}} \right)^{\dfrac{5}{2}}}\].
To do this, we will use the law of exponent \[{\left( {{a^m}} \right)^n} = {a^{mn}}\].
Let us rewrite the exponent \[\dfrac{5}{2}\] using the above law.
We have to express this as a product of two numbers. We write \[\dfrac{5}{2}\] as
\[\dfrac{5}{2} = 5 \times \dfrac{1}{2}\]
Using the law of exponent \[{\left( {{a^m}} \right)^n} = {a^{mn}}\], we will take \[m = \dfrac{1}{2}\] and \[n = 5\].
This means that we will first raise \[\dfrac{1}{2}\] to the exponent \[\dfrac{1}{2}\].
Now, raising a value to the exponent \[\dfrac{1}{2}\] is the same as taking the square root. When we take the square root of \[\dfrac{1}{2}\], two cases arise:
Case 1: The square root of \[\dfrac{1}{2}\] is \[\dfrac{1}{{\sqrt 2 }}\].
Now, we will raise \[\dfrac{1}{{\sqrt 2 }}\] to the exponent \[5\]. This means that we have to multiply \[\dfrac{1}{{\sqrt 2 }}\] by itself five times.
\[{\left( {\dfrac{1}{{\sqrt 2 }}} \right)^5} = \dfrac{1}{{\sqrt 2 }} \times \dfrac{1}{{\sqrt 2 }} \times \dfrac{1}{{\sqrt 2 }} \times \dfrac{1}{{\sqrt 2 }} \times \dfrac{1}{{\sqrt 2 }}\]
Multiplying the terms, we get
So, the value of \[{\left( {\dfrac{1}{2}} \right)^{\dfrac{5}{2}}}\] in this case is \[\dfrac{1}{{4\sqrt 2 }}\].
Case 2: The square root of \[\dfrac{1}{2}\] is \[ - \dfrac{1}{{\sqrt 2 }}\].
Now, we will raise \[ - \dfrac{1}{{\sqrt 2 }}\] to the exponent \[5\]. We can write \[ - \dfrac{1}{{\sqrt 2 }}\] as \[ - \dfrac{1}{{\sqrt 2 }} = ( - 1) \times \dfrac{1}{{\sqrt 2 }}\]. So, when we raise \[ - \dfrac{1}{{\sqrt 2 }}\] to the exponent \[5\], we are actually raising the product of \[ - 1\] and \[\dfrac{1}{{\sqrt 2 }}\] to the exponent \[5\]. We get,
\[{\left( { - \dfrac{1}{{\sqrt 2 }}} \right)^5} = {( - 1)^5} \times {\left( {\dfrac{1}{{\sqrt 2 }}} \right)^5} = - \dfrac{1}{{4\sqrt 2 }}\]
In the above we have used the law of exponent \[{(ab)^m} = {a^m} \times {b^m}\].

So, the value of \[{\left( {\dfrac{1}{2}} \right)^{\dfrac{5}{2}}}\] in this case is \[ - \dfrac{1}{{4\sqrt 2 }}\].

Note:
An alternate method to simplify the value of \[{\left( {\dfrac{1}{2}} \right)^{\dfrac{5}{2}}}\] is as follows:
In the law of exponent \[{\left( {{a^m}} \right)^n} = {a^{mn}}\], let us take \[m = 5\] and \[n = \dfrac{1}{2}\].
This means that we must first raise \[\dfrac{1}{2}\] to the exponent \[5\] and then take the square root.
Now, \[{\left( {\dfrac{1}{2}} \right)^5} = \dfrac{1}{{32}}\]
Taking square root, we get
\[{\left( {\dfrac{1}{{32}}} \right)^{\dfrac{1}{2}}} = \dfrac{1}{{4\sqrt 2 }}\] or \[ - \dfrac{1}{{4\sqrt 2 }}\].