Question

Simplify and express in exponential form
 \[\;\dfrac{{{{({9^3})}^2} \times {7^4}}}{{{3^8} \times 21}}\]

Answer 1

Hint: To simplify the given expression try to break in small power rather than directly calculating the value of the given expression, we basically use laws of exponents to solve such problems.

Complete step-by-step answer:
The given expression is
 \[\;\dfrac{{{{({9^3})}^2} \times {7^4}}}{{{3^8} \times 21}}\]
In numerator multiply the powers of 9 and in denominator break 21 as \[3 \times 7\]
We get,
 \[ = \dfrac{{{{\left( 9 \right)}^{3 \times 2}} \times {7^4}}}{{{3^8} \times 7 \times 3}}\]
In denominator add all the powers of 3
We get,
 \[ = \dfrac{{{9^6} \times {7^4}}}{{{3^9} \times 7}}\]
Bring 7 from denominator to numerator and add the power
We get,
 \[ = \dfrac{{{9^6} \times {7^{4 - 1}}}}{{{3^9}}} = \dfrac{{{9^6} \times {7^3}}}{{{3^9}}}\]
Write 9 as 32 then add their powers
We get,
 \[\; = \dfrac{{{{({3^2})}^6} \times {7^3}}}{{{3^9}}} = \dfrac{{{3^{12}} \times {7^3}}}{{{3^9}}}\]
Bring 39 to the numerator and add their power
We get,
 \[ = {3^{12 - 9}} \times {7^3} = {3^3} \times {7^3}\]
Multiply 3 and 7 as the power is same
 \[{\left( {21} \right)^3}\]
We get 213 so multiply 21 three times
We get,
 \[ = 21 \times 21 \times 21\]
Hence the simplified form is
 \[ = 9261\]
So, the correct answer is “9621”.

Note: 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.