Question

How do you simplify the expression $\dfrac{24{{b}^{2}}}{56b}$?

Answer 1

Hint: Now to solve the given problem we will first factorize the numerator and denominator of the expression by using the definition of indices. Now we will cancel the common terms in numerator and denominator and hence we will simplify the given expression.

Complete step by step solution:
Now first let us understand the concept of indices.
Now take an example ${{4}^{3}}$ . Here the given expression is read as 3 raised to 4. Here 3 is called the power of 4.
Now the power tells us How many times a number is multiplied to itself.
Hence if we have ${{4}^{3}}$ Since the power is 3 this means 4 is multiplied to itself 3 times.
Hence we get, ${{4}^{3}}=4\times 4\times 4$ .
Now the power of a term can be positive, negative or a fraction as well. Now we know how to expand positive powers. Let us see what happens in case of negative powers.
If we have ${{a}^{-m}}$ it can be written as $\dfrac{1}{{{a}^{m}}}$ .
Now if the power of any term is 0 then its value is 1.
 Now consider the given expression $\dfrac{24{{b}^{2}}}{56b}$
Let us factorize the numerator and denominator in the given expression. Hence we get,
$\Rightarrow \dfrac{\left( 2\times 2\times 2\times 3 \right)\left( b\times b \right)}{\left( 2\times 3\times 3\times 3 \right)\left( b \right)}$
Now canceling the common terms in numerator and denominator we get,
$\begin{align}
  & \Rightarrow \dfrac{\left( 2\times 2 \right)\left( b \right)}{\left( 3\times 3 \right)} \\
 & \Rightarrow \dfrac{4b}{9} \\
\end{align}$
Hence we get, $\dfrac{24{{b}^{2}}}{56b}=\dfrac{4b}{9}$

Note: Now note that if there is a fraction in the expression then the fraction can be converted in root form. Hence we have ${{a}^{\dfrac{p}{q}}}=\sqrt[q]{{{a}^{p}}}$ . Hence using this we can also simplify indices with powers as fractions.