Question

How do you use the law of exponents to simplify ${({3^{\dfrac{5}{4}}})^{\dfrac{2}{9}}}$ ?

Answer 1

Hint: When a number is multiplied with itself “n” times, the number is raised to the power “n”, for example, let the number be “a” that is multiplied with itself “n” times, it can be written as ${a^n}$ . In this question, 3 is raised to the power $\dfrac{5}{4}$ , that is, 3 is multiplied with itself $\dfrac{5}{4}$ times, then the number obtained is raised to the power $\dfrac{2}{9}$ , that is, the number obtained is multiplied with itself $\dfrac{2}{9}$ times. The exponential functions obey certain rules called the laws of exponents. Using one of these laws we have to simplify the given exponential function.

Complete step-by-step solution:
The law of exponents that will be used here states that when a quantity is raised to some power and this whole quantity is again raised to some power, then keeping the base quantity same, we multiply the two powers, that is, ${({a^x})^y} = {a^{x \times y}}$ .
So, ${({3^{\dfrac{5}{4}}})^{\dfrac{2}{9}}}$ is equal to ${3^{\dfrac{5}{4} \times }}^{\dfrac{2}{9}}$
$ \Rightarrow {({3^{\dfrac{5}{4}}})^{\dfrac{2}{9}}} = {3^{\dfrac{5}{{18}}}}$
Hence, the simplified form of ${({3^{\dfrac{5}{4}}})^{\dfrac{2}{9}}}$ is ${3^{\dfrac{5}{{18}}}}$ .

Note: In this question, we are given an equation that is in exponential form. When a number is raised to some power, the number is said to be in exponential form, for example ${a^x}$ is an exponential function. There are several laws of exponents like the addition of two exponential functions, subtraction of two exponential functions, etc. The necessary condition for applying these laws is that the base of the exponential functions should be the same. Thus, similar questions can be solved by using the laws of exponents.