Question

How do you simplify \[{{\left( 3a \right)}^{-3}}\] and write it using only positive exponents?

Answer 1

Hint: In order to find solution to this problem, we will solve the question by using the law of exponents which states that when an exponent is raised to a negative power then we reciprocate the base keeping the power the same but dropping the negative sign. This way, we can convert the only negative exponent in the given equation to positive and thus write it using only positive exponents.

Complete step-by-step answer:
Negative exponent that is negative power is defined as the multiplicative inverse of the base raised to the positive opposite of the power.
All negative exponents can be expressed as their positive reciprocal. A reciprocal is a fraction where the numerator and denominator switch places.
$\Rightarrow {{a}^{-n}}=\dfrac{1}{{{a}^{n}}}$
We can solve negative exponents by flipping the base and exponent into the reciprocal, then we will solve the denominator. Then, Divide the numerator by the denominator to find the final decimal.
We are given with monomial to exponent as:
\[\Rightarrow {{\left( 3a \right)}^{-3}}\]
We know that ${{a}^{-n}}=\dfrac{1}{{{a}^{n}}}$,
On applying this formula on our problem, we get:
\[\Rightarrow {{\left( 3a \right)}^{-3}}=\dfrac{1}{{{\left( 3a \right)}^{3}}}\]
Since $3$ in denominator is positive, the obtained expression contains only positive exponents.
Therefore, the simplified form of \[{{\left( 3a \right)}^{-3}}\] is \[\dfrac{1}{{{\left( 3a \right)}^{3}}}\].

Note: Two things to note while solving this problem:
First, a negative exponent always means that the power can be written with a positive exponent by moving that power to the other side of a quotient that is a fraction. So, the \[{{\left( 3a \right)}^{-3}}\] becomes the denominator of a fraction that is the divisor in a quotient .
Second, since the brackets are placed around the entire $3a$ monomial, this will become the base to which the exponent applies and not just the $a$ part.