Question

How do you simplify \[{{\left( -3{{x}^{2}}{{y}^{3}} \right)}^{4}}\]?

Answer 1

Hint: This type of problem can be solved using power rule of multiplication, that is, \[{{\left( ab \right)}^{n}}={{a}^{n}}{{b}^{n}}\]. First, we have to consider the given function. And simplify the given function using the power rule of multiplication. Then, use the rule \[{{\left( {{a}^{n}} \right)}^{m}}={{a}^{nm}}\] for the x and y term. We know that \[{{\left( -3 \right)}^{4}}=81\]. Using these properties, we get a simplified function which is the required answer.

Complete step-by-step solution:
According to the question, we are asked to simplify \[{{\left( -3{{x}^{2}}{{y}^{3}} \right)}^{4}}\].
We have been given the function is \[{{\left( -3{{x}^{2}}{{y}^{3}} \right)}^{4}}\]. ---------(1)
We know the power rule of multiplication, that is, \[{{\left( ab \right)}^{n}}={{a}^{n}}{{b}^{n}}\].
Let us use this power rule in the given function (1).
Here, we have three terms and hence we get,
\[{{\left( -3{{x}^{2}}{{y}^{3}} \right)}^{4}}={{\left( -3 \right)}^{4}}{{\left( {{x}^{2}} \right)}^{4}}{{\left( {{y}^{3}} \right)}^{4}}\] --------(2)
Let us first find the value of the constant.
We know that \[{{\left( 3 \right)}^{4}}=81\].
Therefore, \[{{\left( -3 \right)}^{4}}={{\left( -1 \right)}^{4}}81\].
Since \[{{\left( -1 \right)}^{4}}=1\], we get
\[{{\left( -3 \right)}^{4}}=81\]
Now, let us consider \[{{\left( {{x}^{2}} \right)}^{4}}\].
We know that \[{{\left( {{a}^{n}} \right)}^{m}}={{a}^{nm}}\]. Using this property of powers in the above function, we get
\[{{\left( {{x}^{2}} \right)}^{4}}={{x}^{2\times 4}}\]
On further simplification, we get
\[{{\left( {{x}^{2}} \right)}^{4}}={{x}^{8}}\]
Now, consider \[{{\left( {{y}^{3}} \right)}^{4}}\].
We know that \[{{\left( {{a}^{n}} \right)}^{m}}={{a}^{nm}}\]. Using this property of powers in the above function, we get
\[{{\left( {{y}^{3}} \right)}^{4}}={{y}^{3\times 4}}\]
On further simplification, we get
\[{{\left( {{y}^{3}} \right)}^{4}}={{y}^{12}}\]
Now, substitute all the simplified expressions, that is, 81, \[{{x}^{8}}\] and \[{{y}^{12}}\] in equation (2).
Therefore, \[{{\left( -3 \right)}^{4}}{{\left( {{x}^{2}} \right)}^{4}}{{\left( {{y}^{3}} \right)}^{4}}=81{{x}^{8}}{{y}^{12}}\].
Let us now compare this with equation (1).
Therefore, \[{{\left( -3{{x}^{2}}{{y}^{3}} \right)}^{4}}=81{{x}^{8}}{{y}^{12}}{{y}^{12}}\].
Hence, the simplified form of the function \[{{\left( -3{{x}^{2}}{{y}^{3}} \right)}^{4}}\] is \[81{{x}^{8}}{{y}^{12}}{{y}^{12}}\].

Note:We should not forget to split the terms at the very first. Also avoid calculation
mistakes based on sign conventions. We should use the properties of power to solve this
type of questions without which we cannot find the final answer. Similarly, we can simplify
the functions with more variables also.