Question

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

Answer 1

Hint: In the given question, we are required to simplify an algebraic expression given to us in the problem. So, we have to open and solve the brackets first so as to simplify the algebraic expression. We can use algebraic rules and properties in order to simplify the given algebraic expression. We should keep in mind the order in which the algebraic operations are to be executed so as to obtain the correct answer. The given problem also requires knowledge of some basic exponents and powers properties.

Complete step-by-step answer:
We would use the BODMAS rule in order to simplify the algebraic expression $ {\left( {3{k^4}} \right)^4} $ . BODMAS is an acronym for the sequence in which the mathematical operations are to be done. In BODMAS, B stands for brackets, O stands for of, D stands for division, M stands for multiplication, A stands addition, S stands subtraction.
Therefore, we have, $ {\left( {3{k^4}} \right)^4} $
We know the law of exponents according to which $ {\left( {xy} \right)^a} $ can be written as $ {x^a}{y^a} $ .
So, applying the law of exponents on the expression we have, we get,
 $ \Rightarrow {3^4} \times {\left( {{k^4}} \right)^4} $
Now, we know that the expression $ {\left( {{a^x}} \right)^y} $ is equivalent to $ {a^{xy}} $ . So, we get,
 $ \Rightarrow {3^4} \times {k^{4 \times 4}} $
Simplifying the calculations further, we get,
 $ \Rightarrow {3^4} \times {k^{16}} $
We know that $ {3^4} = 81 $ . So, we get,
 $ \Rightarrow 81{k^{16}} $
Therefore, $ {\left( {3{k^4}} \right)^4} $ can be simplified as $ 81{k^{16}} $ using the laws of exponents.
So, the correct answer is “ $ 81{k^{16}} $”.

Note: Laws of Exponents. When multiplying like bases, keep the base the same and add the exponents. When raising a base with a power to another power, keep the base the same and multiply the exponents. When dividing like bases, keep the base the same and subtract the denominator exponent from the numerator exponent.