Question

Simplify and write in exponential form with negative exponent
\[5^{3} \times \left( \dfrac{4}{5} \right)^{3}\]

Answer 1

Hint: In this question, we need to simplify and write the given exponential in the form of a negative exponent. The given exponential is \[5^{3} \times \left( \dfrac{4}{5} \right)^{3}\]. With the help of the law of exponent we can simplify and write in the form of negative exponent. Negative exponent is that the power of a number is negative. The law of exponent is nothing but it is used to simplify the multiplication and division operations and helps to solve the problem easily.

Complete step by step answer:
Given,
\[5^{3} \times \left( \dfrac{4}{5} \right)^{3}\]
We need to first simplify the given exponential.
According to law of exponent,
\[\ \left( \dfrac{a}{b} \right)^{n} = \dfrac{a^{n}}{b^{n}}\]
By rewriting the given,
We get,
⇒ \[5^{3} \times \left( \dfrac{4^{3}}{5^{3}} \right)\]
Again by the law of exponent,
\[\dfrac{a^{m}}{a^{n}} = \ a^{m – n}\]
Thus we can rewrite as,
⇒ \[\ 5^{(3 – 3)}\ \times 4^{3}\]
By subtracting the powers,
We get,
⇒ \[5^{0} \times 4^{3}\]
Again by the law of exponent,
\[a^{0} = 1\]
By substituting,
We get,
⇒ \[1 \times 4^{3}\]
By law of exponent,
\[a^{n} = \dfrac{1}{a^{- n}}\]
Thus we can write \[4^{3}\] as \[\dfrac{1}{4^{- 3}}\]
⇒ \[1 \times \dfrac{1}{4^{- 3}}\]
By multiplying,
We get,
⇒ \[\dfrac{1}{4^{- 3}}\]
We know that \[1\] to the power any number is again \[1\] , thus we can
Rewrite \[1\] as \[1^{- 3}\]
⇒ \[\dfrac{{(1)}^{- 3}}{{(4)}^{- 3}}\]
By the law of exponent,
\[\dfrac{a^{m}}{a^{n}} = \ a^{m – n}\]
We can rewrite as,
⇒ \[\left( \dfrac{1}{4} \right)^{- 3}\]
Thus we have expressed it in the form of a negative exponent.

Note:
Exponents are nothing but it helps to show the repeated multiplication of a number in the power of a number. That is mathematically it denotes the number of times that a number needs to be multiplied . For example, \[4^{3}\] that is a number \[4\] needs to be multiplied three times. \[4^{3} = 4 \times 4 \times 4\] . The law of exponent is applicable only if the base of the exponent is the same or the powers of the exponent.