Question

How do you evaluate ${{\left( \dfrac{3}{4} \right)}^{3}}$ ?

Answer 1

Hint: Here in this question, we are given different bases with the same power and both are dividing each other. The rules of different bases and same power will be applied: ${{\left( \dfrac{a}{b} \right)}^{x}}=\dfrac{{{a}^{x}}}{{{b}^{x}}}$. Also, solutions will be expressed in positive exponents only and if in case powers become negative, then we have to make them positive before reaching the final answer. You should be familiar with all the properties of exponents and powers.

Complete step by step answer:
Now, let’s solve the question.
We know that exponents can be expressed in the form: ${{a}^{x}}$. It can be read as ‘a’ raise to the power ‘x’ where ‘a’ is the base and ‘x’ is the power. You have to keep in mind that the value of ‘a’ should be greater than zero and cannot be equal to one and the value of ‘x’ can be a real number.
Now, let’s discuss some important functions for exponents.
$\begin{align}
  & \Rightarrow {{a}^{x}}\times {{a}^{y}}={{a}^{x+y}} \\
 & \Rightarrow \dfrac{{{a}^{x}}}{{{a}^{y}}}={{a}^{x-y}} \\
 & \Rightarrow {{\left( {{a}^{x}} \right)}^{y}}={{a}^{xy}} \\
 & \Rightarrow {{a}^{x}}\times {{b}^{x}}={{\left( ab \right)}^{x}} \\
 & \Rightarrow \dfrac{{{a}^{x}}}{{{b}^{x}}}={{\left( \dfrac{a}{b} \right)}^{x}} \\
 & \Rightarrow {{a}^{0}}=1 \\
 & \Rightarrow {{a}^{-x}}=\dfrac{1}{{{a}^{x}}} \\
\end{align}$
Write the expression given in question:
$\Rightarrow {{\left( \dfrac{3}{4} \right)}^{3}}$
As we know from the properties of exponents:
$\Rightarrow {{\left( \dfrac{a}{b} \right)}^{x}}=\dfrac{{{a}^{x}}}{{{b}^{x}}}$
When we apply this property in our expression, we will get:
$\Rightarrow \dfrac{{{3}^{3}}}{{{4}^{3}}}$
Power is 3 means we have to multiply numerator and denominator 3 times. We will get:
$\Rightarrow \dfrac{3\times 3\times 3}{4\times 4\times 4}$
Now, solve further:
$\therefore \dfrac{27}{64}$
So this is the final answer.

Note: There is a shortcut method to solve this expression directly without applying any property. Let’s see that also. First write the given expression:
$\Rightarrow {{\left( \dfrac{3}{4} \right)}^{3}}$
As the power is 3, so multiply the whole fraction 3 times. We will get:
$\Rightarrow \left( \dfrac{3}{4} \right)\times \left( \dfrac{3}{4} \right)\times \left( \dfrac{3}{4} \right)$
Cube of 3 is 27 and cube of 4 is 64. So we will get:
$\Rightarrow \left( \dfrac{27}{64} \right)$
Must remember all the properties of exponents and powers.