Question

How do you simplify ${{\left( 3x \right)}^{-1}}$?

Answer 1

Hint: To simplify the expression given in the question, which is ${{\left( 3x \right)}^{-1}}$, we have to use the properties of the exponents. Firstly, we have to use the property of the negative exponent. The property or the rule of the negative exponent states that when the negative exponents in the numerator are shifted to the denominator, they become positive exponents, that is, ${{a}^{-m}}=\dfrac{1}{{{a}^{m}}}$. And then, we have to use the property of the exponent of a product. The property of the exponent of a product states that the product of the two terms raised to a power is equal to the product of the each term raised to the same power, that is, ${{\left( ab \right)}^{m}}={{a}^{m}}{{b}^{m}}$.

Complete step-by-step answer:
Let us write the expression given in the above question as
$\Rightarrow E={{\left( 3x \right)}^{-1}}$
From the negative exponent rule we know that when the negative powers in the numerator are shifted to the denominator, they become positive powers. So we can write the above expression as
$\Rightarrow E=\dfrac{1}{{{\left( 3x \right)}^{1}}}$
Now, from the exponent of the product rule, we know that the product of the two terms raised to a power is equal to the product of the each term raised to the same power. So we can write the above expression as
$\Rightarrow E=\dfrac{1}{{{3}^{1}}{{x}^{1}}}$
Now, we know that a number raised to the power of one is equal to the number itself. So we can put ${{3}^{1}}=3$ and ${{x}^{1}}=x$ in the above expression to get
$\Rightarrow E=\dfrac{1}{3x}$
Hence, the simplified form of the given expression ${{\left( 3x \right)}^{-1}}$ is equal to $\dfrac{1}{3x}$.

Note: We may not apply the exponent of the product rule in the above solution since we can consider the power of one on the product $\left( 3x \right)$ in the expression $\dfrac{1}{{{\left( 3x \right)}^{1}}}$ and write it simply as $\dfrac{1}{3x}$.