Question

How do you multiply \[{{9}^{8}}\times {{\left( {{9}^{10}} \right)}^{-4}}\]?

Answer 1

Hint: In order to solve the given expression i.e. \[{{9}^{8}}\times {{\left( {{9}^{10}} \right)}^{-4}}\], first we need to apply the law of exponents i.e. \[{{\left( {{a}^{m}} \right)}^{n}}={{a}^{m\times n}}\]. Then simplifying the expression, we need to use the laws of exponent which states that from the law of product with the same base that while multiplying the exponential terms with the same base, then we will add the exponent’s i.e. \[{{a}^{m}}\times {{a}^{n}}={{a}^{m+n}}\]. Then applying this law of exponent and simplifying the expression, we will get the required answer of the given question.

Formula used:
\[{{\left( {{a}^{m}} \right)}^{n}}={{a}^{m\times n}}\]
\[{{a}^{m}}\times {{a}^{n}}={{a}^{m+n}}\]

Complete step by step answer:
We have given that,
\[\Rightarrow {{9}^{8}}\times {{\left( {{9}^{10}} \right)}^{-4}}\]
Simplifying the above using the laws of exponents and powers.
We know that,
From the law of power of a power that when we raise a base with a power to another power, we need to keep the base the same and multiply the powers i.e. \[{{\left( {{a}^{m}} \right)}^{n}}={{a}^{m\times n}}\]
Therefore,
\[\Rightarrow {{9}^{8}}\times {{\left( {{9}^{10}} \right)}^{-4}}={{9}^{8}}\times {{9}^{10\times \left( -4 \right)}}\]
Multiplying the exponent of the term, we will get
\[\Rightarrow {{9}^{8}}\times {{9}^{-40}}\]
Now,
As we know that,
From the law of product with the same base that while multiplying the exponential terms with the same base, then we will add the exponent’s i.e. \[{{a}^{m}}\times {{a}^{n}}={{a}^{m+n}}\].
Therefore,
\[\Rightarrow {{9}^{8}}\times {{9}^{-40}}={{9}^{8+\left( -40 \right)}}={{9}^{-32}}\]
As we know that,
We can write the base i.e. equal to 9 as,
\[9=3\times 3={{3}^{2}}\]
Therefore,
The above expression can be rewritten as,
\[\Rightarrow {{9}^{-32}}={{\left( {{3}^{2}} \right)}^{-32}}\]
Again using the property of laws and exponent i.e. \[{{\left( {{a}^{m}} \right)}^{n}}={{a}^{m\times n}}\]
We will get,
\[\Rightarrow {{9}^{-32}}={{3}^{2\times \left( -32 \right)}}={{3}^{-64}}\]
Therefore,
\[\Rightarrow {{9}^{8}}\times {{\left( {{9}^{10}} \right)}^{-4}}={{3}^{-64}}\]
Hence, this is the required answer.

Note: Here, we note that when we say \[{{a}^{m}}\], here ‘a’ and ‘m’ cannot be zero at the same time. While solving these types of questions, the conceptual knowledge about laws of exponents and powers is required. Students should always keep in mind the various laws of exponents and powers in order to solve these types of questions.