Question

How do you simplify $\dfrac{7{{x}^{4}}{{y}^{3}}}{5xy}.\dfrac{2x{{y}^{7}}}{21{{y}^{5}}}$ ?

Answer 1

Hint: To solve these types of questions which involve the multiplication of fractions, just cancel out any like terms from the denominator and numerator and then simply multiply all the values in the numerator and the denominator to get the final answer.

Complete step by step solution:
Given the expression:
$\dfrac{7{{x}^{4}}{{y}^{3}}}{5xy}.\dfrac{2x{{y}^{7}}}{21{{y}^{5}}}$
To simplify the given expression, we will use the laws of exponents that states that $\dfrac{{{a}^{m}}}{{{a}^{n}}}={{a}^{m-n}}$ and ${{a}^{m}}{{a}^{n}}={{a}^{m+n}}$.
Simplifying and re-writing the above given expression by using the above mentioned law of exponents, we get,
Here we first use the Law of exponents for multiplication.
$\Rightarrow \dfrac{7{{x}^{4}}{{y}^{3}}}{5xy}.\dfrac{2x{{y}^{7}}}{21{{y}^{5}}}=\dfrac{7\times 2\left( {{x}^{4+1}} \right)\left( {{y}^{3+7}} \right)}{5\times 21\left( x \right)\left( {{y}^{1+5}} \right)}$
Simplifying the above expression by adding the powers of the variables,
$\Rightarrow \dfrac{7{{x}^{4}}{{y}^{3}}}{5xy}.\dfrac{2x{{y}^{7}}}{21{{y}^{5}}}=\dfrac{7\times 2\times {{x}^{5}}\times {{y}^{10}}}{5\times 21\times x\times {{y}^{6}}}$
Again, now use the law of exponents for the division to further simplify the above expression,
$\Rightarrow \dfrac{7\times 2\times {{x}^{5}}\times {{y}^{10}}}{5\times 21\times x\times {{y}^{6}}}$
$\Rightarrow \dfrac{7\times 2\times {{x}^{5-1}}\times {{y}^{10-6}}}{5\times 21}$
Simplify the powers of the variable, we get,
$\Rightarrow \dfrac{7\times 2\times {{x}^{5-1}}\times {{y}^{10-6}}}{5\times 21}$
$\Rightarrow \dfrac{7\times 2\times {{x}^{4}}\times {{y}^{4}}}{5\times 21}$
Now, in the above expression, $21$ gets cancelled by $7$ since it is a multiple of $7$. Therefore, on further simplifying and solving, we get,
$\Rightarrow \dfrac{7\times 2\times {{x}^{4}}\times {{y}^{4}}}{5\times 21}=\dfrac{2{{x}^{4}}{{y}^{4}}}{15}$
Hence, on simplifying the expression given in the question, we get the final answer as $\dfrac{2{{x}^{4}}{{y}^{4}}}{15}$

Note: A fraction can be defined as a representation of a part of a whole. A general fraction includes a numerator and a denominator which is non-zero. The numerator and denominator are separated by a slash or a line. Apart from general or common fractions, there are many other types of fractions as well, which include compound fractions, complex fractions, and mixed fractions.
To multiply any two fractions, just multiply all the values of numerators and the denominators and then proceed to further simplify the fraction if needed. Sometimes only a single whole number is given in place of a fraction for multiplication, in such a case treating the whole number as a fraction itself by taking the denominator as $1$ .