Question

How do you simplify \[{(\dfrac{3}{y})^4}\] ?

Answer 1

Hint: We are given a fraction that is raised to some power. We know that exponential functions are distributive, that is, ${(a)^x} = {a^x}$ so we will write the numerator and the denominator raised to the power. A number raised to some power “n” signifies that the number is multiplied with itself “n” times. So, according to the power, we will simplify the numerator and the denominator. This way we will get the simplified value of the given exponential function.

Complete step-by-step solution:
We have to simplify \[{(\dfrac{3}{y})^4}\]
We know that exponential functions are distributive, so –
\[{(\dfrac{3}{y})^4} = \dfrac{{{3^4}}}{{{y^4}}}\]
As both the numerator and the denominator are raised to the power 4, it means that they both are multiplied with themselves 4 times. So, we will write ${3^4}$ as $3 \times 3 \times 3 \times 3$ but y is an unknown quantity so it cannot be simplified further.
\[{(\dfrac{3}{y})^4} = \dfrac{{3 \times 3 \times 3 \times 3}}{{{y^4}}}\]
The product of 3 multiplied with itself 4 times is 81.
$ \Rightarrow {(\dfrac{3}{y})^4} = \dfrac{{81}}{{{y^4}}}$
Hence, the simplified form of \[{(\dfrac{3}{y})^4}\] is $\dfrac{{81}}{{{y^4}}}$

Note: In this question, we are given a fraction raised to some power. A fraction is defined as an expression in which two numerical values are divided into two parts by a horizontal line. The part above the horizontal line is known as the numerator and the part below the horizontal line is known as the denominator. In the given fraction 3 is the numerator and y is the denominator. The expression is an algebraic expression as it is a combination of numerical values and alphabets where they both are linked by an arithmetic operation (division).