Question

How do you simplify ${\left( {2{a^3}b} \right)^5}?$

Answer 1

Hint: To simplify the given expression, go through the BODMAS rule, first simplify any algebraic operation present under the bracket then simplify the orders of the expression that is exponent. To simplify the exponents use law of indices for brackets and also distributive property of indices over multiplication to simplify the given expression.
Law of indices for brackets can be understood as follows
${\left( {{x^m}} \right)^n} = {x^{m \times n}}$
And distributive property of indices over multiplication can be understood as
${\left( {x \times y} \right)^m} = {x^m} \times {y^m}$

Complete step by step solution:
In order to simplify the given expression ${\left( {2{a^3}b} \right)^5}$ we will use distributive property of exponents over multiplication, which is given as if there is an exponent on product of two or more numbers or variables then we can write it as product of exponent on each number and can be understood mathematically as
${\left( {x \times y} \right)^m} = {x^m} \times {y^m}$
Using this we can write the given expression as
\[
   \Rightarrow {\left( {2{a^3}b} \right)^5} \\
    \Rightarrow {2^5}{\left( {{a^3}} \right)^5}{b^5} \\
 \]
Now, to further simplify we will use the law of indices for brackets which says that if a number is raised to the power of an exponent and again that exponential number is raised to the power of another exponent then it is simplified as the number raises to the power product of both exponents. Mathematically it is given as
${\left( {{x^m}} \right)^n} = {x^{m \times n}}$
Using this simplify the expression further, we will get
\[
    \Rightarrow {2^5}{\left( {{a^3}} \right)^5}{b^5} \\
    \Rightarrow {2^5}{a^{3 \times 5}}{b^5} \\
    \Rightarrow {2^5}{a^{15}}{b^5} \\
    \Rightarrow 32{a^{15}}{b^5} \\
 \]
Therefore \[32{a^{15}}{b^5}\] is the simplified form of ${\left( {2{a^3}b} \right)^5}$

Note: Apart from bracket, law of indices is used in multiplication, division, fraction power, negative power and zero power. Law of indices is being used to simplify problems having exponents in it, and understanding it will help you in simplifying this type of problems.