Question

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

Answer 1

Hint: Here we are asked to simplify the exponents. Firstly, we make use of power of a fraction rule given by ${\left( {\dfrac{a}{b}} \right)^m} = \dfrac{{{a^m}}}{{{b^m}}}$. Then use the product rule which is given by ${(ab)^m} = {a^m}{b^m}$. After that we use power of a power rule of exponents which is given by ${({a^m})^n} = {a^{mn}}$ and simplify. Then we solve using the division rule of exponents $\dfrac{{{a^m}}}{{{a^n}}} = {a^{m - n}}$ to obtain the required result.

Complete step by step solution:
Here we are given ${\left( {\dfrac{{{x^2}{y^3}}}{{x{y^3}}}} \right)^2}$.
Now we distribute the power outside the bracket to both numerator and denominator.
We use the power of a fraction rule given by ${\left( {\dfrac{a}{b}} \right)^m} = \dfrac{{{a^m}}}{{{b^m}}}$
Hence we can also write the above expression as,
$\dfrac{{{{({x^2}{y^3})}^2}}}{{{{(x{y^3})}^2}}}$
We need to simplify the above exponents using some rules of exponents
Now we use the power of a product rule given by ${(ab)^m} = {a^m}{b^m}$.
Here $m = 2$. Applying the power of a product rule to both numerator and denominator we get,
$ \Rightarrow \dfrac{{{{({x^2})}^2}{{({y^3})}^2}}}{{{x^2}{{({y^3})}^2}}}$
Firstly, we multiply the power outside the brackets. We use the power of a power rule of exponents to do this.
The power of a power rule of exponents is given by ${({a^m})^n} = {a^{mn}}$.
So using this we get,
$ \Rightarrow \dfrac{{{x^{2 \times 2}}{y^{3 \times 2}}}}{{{x^2}{y^{3 \times 2}}}}$
Multiplying the powers of exponents we get,
$ \Rightarrow \dfrac{{{x^4}{y^6}}}{{{x^2}{y^6}}}$
Now we observe that in both numerator and denominator ${y^6}$ is common. We canceled this term to get the solution.
So, we use the division rule of exponents, which is given by
$\dfrac{{{a^m}}}{{{a^n}}} = {a^{m - n}}$
Note that here $m = 6$ and $n = 6$.
Hence applying division rule of exponents to the term y we get,
$ \Rightarrow \dfrac{{{x^4}{y^{6 - 6}}}}{{{x^2}}}$
$ \Rightarrow \dfrac{{{x^4}{y^0}}}{{{x^2}}}$
We know that ${a^0} = 1$. Therefore ${y^0} = 1.$
$ \Rightarrow \dfrac{{{x^4} \times 1}}{{{x^2}}}$
$ \Rightarrow \dfrac{{{x^4}}}{{{x^2}}}$
Now again we use the division rule of exponents.
Here $m = 4$and $n = 2$. Hence we get,
$ \Rightarrow {x^{4 - 2}}$
$ \Rightarrow {x^2}$

Hence the simplification of the term ${\left( {\dfrac{{{x^2}{y^3}}}{{x{y^3}}}} \right)^2}$is ${x^2}$.

Note :
Students must remember the rules of exponents to simplify such problems. We need to be careful while applying the rules. It's 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 : ${a^{ - x}} = \dfrac{1}{{{a^x}}}$