Question

How do you simplify ${\left( {2{a^3}} \right)^4}{\left( {{a^3}} \right)^3}?$

Answer 1

Hint: Use BODMAS rule, first solve the orders in the equation with help of distributive property of exponent over multiplication and then with help of law of indices for brackets, then secondly multiply the terms present in the expression with help of law of indices for multiplication.
Law of indices for brackets can be understood as follows
${\left( {{x^m}} \right)^n} = {x^{m \times n}}$
Law of indices for multiplication can be understood as follows
\[{x^m} \times {x^n} = {x^{m + n}}\]
And distributive property of indices over multiplication can be understood as
${\left( {x \times y} \right)^m} = {x^m} \times {y^m}$

Formula used:
Law of indices for brackets: ${\left( {{x^m}} \right)^n} = {x^{m \times n}}$
Law of indices for multiplication: \[{x^m} \times {x^n} = {x^{m + n}}\]
Distributive property of exponent over multiplication: ${\left( {x \times y} \right)^m} = {x^m} \times {y^m}$

Complete step-by-step solution:
In order to simplify the given expression ${\left( {2{a^3}} \right)^4}{\left( {{a^3}} \right)^3}$ we will go through BODMAS rule, and according to BODMAS rule orders will be first simplified in the expression and then multiplication, so first simplifying its terms with help of distributive property of exponent over multiplication and law of indices for brackets as follows
\[ = {\left( {2{a^3}} \right)^4}{\left( {{a^3}} \right)^3}\]
Using distributing property of exponent over multiplication, we will get
$ = \left( {{2^4}{{\left( {{a^3}} \right)}^4}} \right){\left( {{a^3}} \right)^3}$
Now, we will use law of indices for brackets to simplify its terms further,
$
   = \left( {{2^4}{a^{3 \times 4}}} \right)\left( {{a^{3 \times 3}}} \right) \\
   = \left( {16{a^{12}}} \right)\left( {{a^9}} \right) \\
 $
So we have simplified the terms, now to simplify the expression, multiplying them with help of law of indices for multiplication, we will get
$
   = 16{a^{12}} \times {a^9} \\
   = 16{a^{12 + 9}} \\
   = 16{a^{21}} \\
 $
Therefore $16{a^{21}}$ is the simplified version of the given expression.


Note: Here to simplify the given question we use distribution property. Distribution property of exponent holds good only for multiplication and division, for addition and subtraction of two or more numbers we use binomial expansion to distribute exponent or open the parentheses among the terms.