Question

How do you simplify \[{4^{\dfrac{5}{2}}}\]?

Answer 1

Hint: We will first rewrite the exponent of the given number as a product of two numbers using the laws of the exponent. We will then apply each exponent separately on the term and simplify it 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 \[{4^{\dfrac{5}{2}}}\].
To do this, we will use the law of exponent \[{\left( {{a^m}} \right)^n} = {a^{mn}}\].
We will 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 \[4\] 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 \[4\], two cases arise:
Case 1: The square root of \[4\] is \[2\].
Now, we will raise \[2\] to the exponent \[5\]. This means that we have to multiply \[2\] by itself five times.
\[{2^5} = 2 \times 2 \times 2 \times 2 \times 2 = 32\]\[\]
So, the value of \[{4^{\dfrac{5}{2}}}\] in this case is \[32\].
Case 2: The square root of \[4\] is \[ - 2\].
Now, we will raise \[ - 2\] to the exponent \[5\]. We can write \[ - 2\] as \[ - 2 = ( - 1) \times 2\]. So, when we raise \[ - 2\] to the exponent \[5\], we are actually raising the product of \[ - 1\] and \[2\] to the exponent \[5\]. We get,
\[{( - 2)^5} = {( - 1)^5} \times {2^5} = - 32\]
In the above we have used the law of exponent \[{(ab)^m} = {a^m} \times {b^m}\].

So, the value of \[{4^{\dfrac{5}{2}}}\] in this case is \[ - 32\].

Note: An alternate method to simplify the value of \[{4^{\dfrac{5}{2}}}\] is as follows:
In the law of exponent \[{\left( {{a^m}} \right)^n} = {a^{mn}}\], let us take \[m = 5\] and \[n = \dfrac{1}{2}\].