Question

How do you simplify \[{25^{\dfrac{3}{2}}}\]?

Answer 1

Hint: Here, we will convert the fractional exponent of a given number into a Radical form, and by using the Radical form, we will simplify the given expression. If the number has a fractional exponent, then the numerator of a fractional exponent represents the power of the radical form and the denominator of a fractional exponent represents the index of the radical form.

Complete Step by Step Solution:
We are given a number \[{25^{\dfrac{3}{2}}}\].
Now, we will convert the fractional exponent into the radical form to simplify the given number. So, we get
\[{25^{\dfrac{3}{2}}} = {\left( {\sqrt[2]{{25}}} \right)^3}\]
Now, to simplify the given number, we will find the square of the number by finding the factors by using the method of Prime Factorization.
We will first divide 25 by the least prime number 5, therefore, we get
\[25 \div 5 =5\]
As the obtained quotient is a prime number, we will not divide the number further. So,
\[25 = 5 \times 5\]
The factors are grouped in pairs to find the square root. Thus, we get
\[ \Rightarrow {25^{\dfrac{3}{2}}} = {\left( {\sqrt[2]{{5 \times 5}}} \right)^3}\]
Now, by simplifying the equation, we get
\[ \Rightarrow {25^{\dfrac{3}{2}}} = {\left( 5 \right)^3}\]
Applying the exponent on the terms, we get
\[ \Rightarrow {25^{\dfrac{3}{2}}} = 125\].

Therefore, the expression \[{\left( {25} \right)^{\dfrac{3}{2}}}\] is simplified to an integer \[125\].

Additional information:
We can write power at first and then the index or we can write index at first and then the power. This can be written in either way to express the number in the Radical form. In simple, we write the fractional exponent of a number \[{x^{\dfrac{p}{q}}}\] in the Radical form as \[\sqrt[q]{{{x^p}}}\] or \[{\left( {\sqrt[q]{x}} \right)^p}\]. The radical form can be written only with the Radical sign.

Note:
We should note that the radical sign indicates to find the root of a number. We know that the radical sign with a small number \[n\] is known as \[{n^{th}}\] root of a number. The smaller number is called the index. We should also remember that usually the square root of a number is not written with any index. We should remember that the number with a fractional exponent into a Radical to simplify the expression at an ease.