Question

Simplify the following:
$3{{\left( {{a}^{4}}{{b}^{3}} \right)}^{10}}\times 5{{\left( {{a}^{2}}{{b}^{2}} \right)}^{3}}$

Answer 1

Hint: To simplify the above expression, we are going to use the property of the exponents which says: ${{\left( {{p}^{x}}{{q}^{y}} \right)}^{z}}={{p}^{xz}}{{q}^{yz}}$. Along with this property, we are going to use the property which says that if the base is same and are in multiplication with each other then the exponents will get added up and the mathematical expression will look like: ${{q}^{m}}\times {{q}^{n}}={{q}^{m+n}}$. Using these properties will help us in simplifying the above expression.

Complete step by step answer:
The expression given in the above problem which we have to simplify is as follows:
$3{{\left( {{a}^{4}}{{b}^{3}} \right)}^{10}}\times 5{{\left( {{a}^{2}}{{b}^{2}} \right)}^{3}}$
First of all, we are going to use the property of the exponents which says that:
${{\left( {{p}^{x}}{{q}^{y}} \right)}^{z}}={{p}^{xz}}{{q}^{yz}}$
The above property we are going to use by simplifying the $a\And b$ written inside the brackets with exponent as 10 and 3. In the below, we are going to show the terms in the above expression where we are going to use this property.
$\begin{align}
  & {{\left( {{a}^{4}}{{b}^{3}} \right)}^{10}}={{a}^{4\times 10}}{{b}^{3\times 10}}={{a}^{40}}{{b}^{30}} \\
 & {{\left( {{a}^{2}}{{b}^{2}} \right)}^{3}}={{a}^{2\times 3}}{{b}^{2\times 3}}={{a}^{6}}{{b}^{6}} \\
\end{align}$
Now, substituting the above simplification in the given expression and we get,
$3{{a}^{40}}{{b}^{30}}\times 5{{a}^{6}}{{b}^{6}}$
Multiplying 3 by 5 we get,
$15{{a}^{40}}{{b}^{30}}{{a}^{6}}{{b}^{6}}$
After that, we are going to use the property which says that when base is same and the two same numbers having some exponents are multiplied to each other then in the result of this multiplication, the exponents of the base will get added up and we get,
${{q}^{m}}\times {{q}^{n}}={{q}^{m+n}}$
Applying the above property, first of all on $''a''$ and then $''b''$we get,
$\begin{align}
  & 15{{a}^{40+6}}{{b}^{30+6}} \\
 & =15{{a}^{46}}{{b}^{36}} \\
\end{align}$

Hence, we have simplified the given expression to: $15{{a}^{46}}{{b}^{36}}$.

Note: To solve the above problem, you must know the exponent properties of the algebraic multiplication of numbers otherwise it would be hard for you to solve this problem. Also, the other exponent property you might need in the other simplification is that if we have to same numbers with different or same exponents are in the quotient form then result of this division is as follows:
$\dfrac{{{s}^{d}}}{{{s}^{e}}}={{s}^{d-e}}$