Question

How do you simplify ${\left( {2x} \right)^{ - 3}}$?

Answer 1

Hint: Here we need to simplify the given mathematical expression. For that, we will use different properties of the exponents. We will first use the distributive property of exponents to distribute the exponents for the terms inside the parenthesis. Then we will use the negative exponent’s rule to get the final answer in which there will be no term with the negative power.

Formula used:
We will use the following formulas:
1. According to the distributive property of exponents:- ${\left( {a \times b} \right)^c} = {a^c} \times {b^c}$
2. Negative power rule of exponents:-${{a}^{-b}}=\dfrac{1}{{{a}^{b}}}$
Here $a$, $b$ and $c$ are real numbers.

Complete step by step solution:
Here we need to simplify the given mathematical expression and the given expression is ${{\left( 2x \right)}^{-3}}$.
We know from the distributive property of exponents that:
${{\left( a\times b \right)}^{c}}={{a}^{c}}\times {{b}^{c}}$
Here $a$, $b$ and $c$ are real numbers.
We will use the same property here in the given expression.
${{\left( 2x \right)}^{-3}} ={{2}^{-3}}\times {{x}^{-3}}$
We know from the Negative power rule of exponents that:
${a^{ - b}} = \dfrac{1}{{{a^b}}}$.
Here $a$ and $b$ are real numbers.
We will use the negative power rule of exponents here for the terms having negative exponents.
$ {{\left( 2x \right)}^{-3}} = \dfrac{1}{8} \times \dfrac{1}{{{x^3}}}$
Now, we will multiply the terms of the expression to further simplify it.

${{\left( 2x \right)}^{-3}} =\dfrac{1}{8{{x}^{3}}}$
Hence, this is the required simplified value of the given expression.

Note:
Here we have obtained the simplified expression using the properties of the exponents. The exponent is defined as the number, which is positioned at the up-right of the base number. We need to remember the basic rules of the exponents. Also, we need to keep in mind that if we multiply the exponents having the same bases, then their exponents get added. Similarly when we divide the exponentials having the same bases, then their exponents get subtracted.