Question

How do you multiply\[-2{{u}^{2}}{{v}^{2}}\times {{\left( {{u}^{4}}{{v}^{4}} \right)}^{3}}\]?

Answer 1

Hint: In the given question, we have been asked to multiply the exponential expression. In order to multiply the given expression, we need to use the law of exponent and powers. First we open the brackets by using the law which states that when raising a base with power to another power, keeping the base the same and multiplying the exponents. Later we simplify the given expression by using a law of exponent which states that when multiplying the like bases, keep the base the same and add the exponents.

Formula used:
The law of exponents of power, which states that when raising a base with power to another power, keeping the base same and multiply the exponents, i.e. \[{{\left( {{a}^{m}} \right)}^{n}}={{a}^{m\times n}}\].
The law of exponents of multiplication, which states that when multiplying the like bases, keep the base same and add the exponents, i.e. \[{{a}^{m}}\times {{a}^{n}}={{a}^{m+n}}\].

Complete step by step solution:
We have given that,
\[\Rightarrow -2{{u}^{2}}{{v}^{2}}\times {{\left( {{u}^{4}}{{v}^{4}} \right)}^{3}}\]
Using the law of exponents of power, which states that when raising a base with power to another power, keeping the base same and multiply the exponents, i.e.
\[{{\left( {{a}^{m}} \right)}^{n}}={{a}^{m\times n}}\]
Tus,
Applying this law in the above given expression, we get
\[\Rightarrow -2{{u}^{2}}{{v}^{2}}\times {{\left( {{u}^{4}} \right)}^{3}}\times {{\left( {{v}^{4}} \right)}^{3}}\]
\[\Rightarrow -2{{u}^{2}}{{v}^{2}}\times \left( {{u}^{4\times 3}} \right)\times \left( {{v}^{4\times 3}} \right)\]
Simplifying the above expression, we get
\[\Rightarrow -2{{u}^{2}}{{v}^{2}}\times {{u}^{12}}\times {{v}^{12}}\]
Rewrite the above expression as;
\[\Rightarrow -2{{u}^{2}}{{v}^{2}}{{u}^{12}}{{v}^{12}}\]
Now,
Using the law of exponents of multiplication, which states that when multiplying the like bases, keep the base same and add the exponents, i.e.
\[{{a}^{m}}\times {{a}^{n}}={{a}^{m+n}}\]
Applying this law of exponent in the above expression, we get
\[\Rightarrow -2{{u}^{2+12}}{{v}^{2+12}}\]
Simplifying the above expression, we get
\[\Rightarrow -2{{u}^{14}}{{v}^{14}}\]
Thus,
\[\Rightarrow -2{{u}^{2}}{{v}^{2}}\times {{\left( {{u}^{4}}{{v}^{4}} \right)}^{3}}=-2{{u}^{14}}{{v}^{14}}\]

Therefore, the product of \[-2{{u}^{2}}{{v}^{2}}\times {{\left( {{u}^{4}}{{v}^{4}} \right)}^{3}}\]is equals to \[-2{{u}^{14}}{{v}^{14}}\]. Hence, it is the required answer.

Note: While solving these types of questions, students need to know about the concepts of multiplication of exponents. They should remember the basic properties of exponential in order to solve the given expression. Students should be very careful while doing calculations to avoid making any errors as it will give us the wrong answer.