Question

How do you solve $x^{\dfrac{3}{2}} = 27$ ?

Answer 1

In this question, we are given an equation in which the left-hand side has the term containing x and the right-hand side contains simple numerical values. We are given a relation in this equation to find the value of x, thus we have to solve either of the sides to convert it into a form similar to the other side of the equation. Then on comparing the two sides of the equation we get the value of x.

Complete step by step answer:
Prime factorization of 27 is done as –
$27 = 3 \times 3 \times 3 \\\ \Rightarrow 27 = 9 \times 3$
Now we know that –
$9 = 3 \times 3 \\\ \Rightarrow 9 = {3^2} \\\ \Rightarrow 3 = {9^{\dfrac{1}{2}}} \\\$
That is 3 is the square root of 9, using this relation in the above-obtained equation, we get –
$27 = {9.9^{\dfrac{1}{2}}} = {9^{1 + \dfrac{1}{2}}} = {9^{\dfrac{3}{2}}}$
On comparing the solved right-hand side with the left-hand side, we get –
{x^{\dfrac{3}{2}}} = {9^{\dfrac{3}{2}}} \\\ \Rightarrow x = 9 \\\
Hence, when ${x^{\dfrac{3}{2}}} = 27$ , $x = 9$ .

Note: There are several laws of exponents that are used to solve the questions involving large powers or for other purposes, as they make the calculations a lot easier. The law we have used here states that when we are multiplying two numbers that have the same base but different exponents, we can keep the base the same and add the exponents, that is, ${a^x}{a^y} = {a^{x + y}}$ . This question can also be solved by another method: we can write ${x^{\dfrac{3}{2}}} = 27$ as that is x is equal to the cube root of the square of 27. We express the square of 27 as a product of its prime factors and write it as a cube of a particular
number to find out the value of x.