Question

How do you simplify (2a^2)^2 ?

Answer 1

Hint: We have first solved all the operations that are inside the bracket , then we can solve outside the bracket . We know the property of exponential function that ${{\left( {{a}^{x}} \right)}^{y}}={{a}^{xy}}$ we can use this formula to simplify the given equation.

Complete step by step answer:
The equation which we have to simplify is (2a^2)^2
We can write this as ${{\left( 2{{a}^{2}} \right)}^{2}}$
We know the exponential formula ${{\left( {{a}^{x}} \right)}^{y}}={{a}^{xy}}$ so and another property of exponential function
${{\left( ab \right)}^{x}}={{a}^{x}}{{b}^{x}}$
Applying this formula to ${{\left( 2{{a}^{2}} \right)}^{2}}$ we get
$\Rightarrow {{\left( 2{{a}^{2}} \right)}^{2}}={{2}^{2}}\times {{\left( {{a}^{2}} \right)}^{2}}$
Square of 2 is equal to 4 so we can write
$\Rightarrow {{\left( 2{{a}^{2}} \right)}^{2}}=4{{\left( {{a}^{2}} \right)}^{2}}$
Now we can apply the formula ${{\left( {{a}^{x}} \right)}^{y}}={{a}^{xy}}$ to solve ${{\left( {{a}^{2}} \right)}^{2}}$ , we can take x is equal to 2 and y is equal to 2 so the value of ${{\left( {{a}^{2}} \right)}^{2}}$ is equal to ${{a}^{4}}$
We can write ${{\left( {{a}^{2}} \right)}^{2}}$ as ${{a}^{4}}$ so the value of ${{\left( 2{{a}^{2}} \right)}^{2}}$ is equal to $4{{a}^{4}}$
So the simplification form of (2a^2)^2 is $4{{a}^{4}}$

Note:
Always note the position of the bracket and solve the operation inside the bracket first. For example if the given question is ${{\left( 2a \right)}^{{{2}^{2}}}}$ then answer would have 2a to the power 4 that is $16{{a}^{4}}$, we have seen if the question is ${{\left( 2{{a}^{2}} \right)}^{2}}$ then the answer is equal to $4{{a}^{4}}$ so position of bracket is very important. Always remember the standard formula for exponential function for example ${{\left( {{a}^{x}} \right)}^{y}}={{a}^{xy}}$ , ${{a}^{x}}{{a}^{y}}={{a}^{x+y}}$ , ${{a}^{x}}{{b}^{x}}={{\left( ab \right)}^{x}}$ , ${{a}^{x-y}}=\dfrac{{{a}^{x}}}{{{a}^{y}}}$ where a is not equal to 0 etc. . These formulas will be very helpful while solving this type of problem