Question

Solve \[ \dfrac{{{7^2} \times {{11}^6} \times 3}}{{{{11}^3} \times 147}}\]

Answer 1

Hint:
Here we will first take the common value in numerator and denominator together and then by using the exponential formula we will simplify it. Then we will find the factor of the denominator value and again cancel the common terms in numerator and denominator and simplify them. Finally we will solve the equation further to get the required answer.

Complete step by step solution:
We have to solve \[\dfrac{{{7^2} \times {{11}^6} \times 3}}{{{{11}^3} \times 147}}\]
We can see that 11 is in the numerator and as well as in the denominator.
So, rewriting the expression, we get
\[ \Rightarrow \dfrac{{{7^2} \times {{11}^6} \times 3}}{{{{11}^3} \times 147}} = \dfrac{{{7^2} \times 3}}{{147}} \times \dfrac{{{{11}^6}}}{{{{11}^3}}}\]
Now using the exponential property \[\dfrac{{{a^m}}}{{{a^n}}} = {a^{m - n}}\], we get
\[ \Rightarrow \dfrac{{{7^2} \times {{11}^6} \times 3}}{{{{11}^3} \times 147}} = \dfrac{{{7^2} \times 3}}{{147}} \times {11^{6 - 3}}\]
\[ \Rightarrow \dfrac{{{7^2} \times {{11}^6} \times 3}}{{{{11}^3} \times 147}} = \dfrac{{{7^2} \times 3}}{{147}} \times {11^3}\]…..\[\left( 1 \right)\]
Next, we will find the factor of the numerator term i.e.147 as,
\[147 = 3 \times 7 \times 7\]
Expressing 7 in terms of exponent, we get
\[ \Rightarrow 147 = 3 \times {7^2}\]
So, replacing the above value in equation \[\left( 1 \right)\], we get,
\[ \Rightarrow \dfrac{{{7^2} \times {{11}^6} \times 3}}{{{{11}^3} \times 147}} = \dfrac{{{7^2} \times 3 \times {{11}^3}}}{{3 \times {7^2}}}\]
Rewriting the above equation, we get
\[ \Rightarrow \dfrac{{{7^2} \times {{11}^6} \times 3}}{{{{11}^3} \times 147}} = \dfrac{{{7^2}}}{{{7^2}}} \times \dfrac{3}{3} \times {11^3}\]
Cancelling the like terms, we get
\[ \Rightarrow \dfrac{{{7^2} \times {{11}^6} \times 3}}{{{{11}^3} \times 147}} = {11^3}\]
Applying the exponent on the terms, we get
\[ \Rightarrow \dfrac{{{7^2} \times {{11}^6} \times 3}}{{{{11}^3} \times 147}} = 1331\]
This is the required answer.

Note:
Exponents and their powers are one of the important topics in mathematics that are used to solve various problems. Some basic notation of exponent and its power is given below:
1) The short notation \[{10^5}\] stands for \[10 \times 10 \times 10 \times 10 \times 10\] where 10 is our base and 4 is its power.
2) We simply call the above value as 10 raised to the power of 5.
3) We add the power of exponents when they are multiplied only if their base is the same.
4) We subtract the power of the exponent when they are divided only if their base is the same.