Question

How do you simplify ${n^3}{({n^3})^3}$?

Answer 1

Hint: In this question, we need to simplify the given exponents. To simplify this, firstly we will evaluate the term inside the parenthesis. To simplify it we use the power of a power rule of exponents which is given by ${({a^m})^n} = {a^{mn}}$. Then we obtain a term of the form ${a^m}{a^n}$. To simplify this we use the product rule of exponents which is given as ${a^m} \cdot {a^n} = {a^{m + n}}$. After that we simplify it and obtain the required form.

Complete step-by-step solution:
Given an exponent of the form ${n^3}{({n^3})^3}$ …… (1) We are asked to simplify the given exponent and find the desired result.
Firstly, we need to evaluate the term in a parenthesis. So we distribute the power outside the parentheses and simplify it.
Consider the term ${({n^3})^3}$.
We will use the power of a power rule to simplify this.
The power of a power rule is given by ${({a^m})^n} = {a^{m \cdot n}} = {a^{mn}}$
Here $m = n = 3$. Hence we get,
$ \Rightarrow {({n^3})^3} = {n^{3 \times 3}}$
$ \Rightarrow {({n^3})^3} = {n^9}$
Hence the equation (1) becomes,
$ \Rightarrow {n^3}{({n^3})^3} = {n^3}({n^9})$
Now this is of the form ${a^m} \cdot {a^n}$.
We use the multiplication rule of exponent which is given by ${a^m} \cdot {a^n} = {a^{m + n}}$.
Here $m = 3$ and $n = 9$. Hence we get,
$ \Rightarrow {n^3}{({n^3})^3} = {n^{3 + 9}}$
$ \Rightarrow {n^3}{({n^3})^3} = {n^{12}}$.
So the simplified form of ${n^3}{({n^3})^3}$ is given by ${n^{12}}$.

Note: Students must remember the rules of exponents to simplify such problems. We need to be careful while applying the rules. It is necessary to use the correct rule to split the terms and simplify the answer.
The rules of exponents are given below.
(1) Multiplication rule : ${a^m} \cdot {a^n} = {a^{m + n}}$
(2) Division rule : $\dfrac{{{a^m}}}{{{a^n}}} = {a^{m - n}}$
(3) Power of a power rule : ${({a^m})^n} = {a^{mn}}$
(4) Power of a product rule : ${(ab)^m} = {a^m}{b^m}$
(5) Power of a fraction rule : ${\left( {\dfrac{a}{b}} \right)^m} = \dfrac{{{a^m}}}{{{b^m}}}$
(6) Zero exponent : ${a^0} = 1$
(7) Negative exponent : ${a^{ - x}} = \dfrac{1}{{{a^x}}}$