Question

Find the value of \[{{11}^{8}}\div \left( {{11}^{10}}\div {{11}^{16}} \right)\]

Answer 1

Hint: The given expression is simplified using laws of exponents. The following law is used:
     \[\dfrac{{{x}^{m}}}{{{x}^{n}}}={{x}^{m-n}}\]
BODMAS is also used for simplification. Hence, the terms within the bracket are simplified first. Rules of addition and subtraction of integers is used. Also, rules of multiplication of signs are also used.

Complete step-by-step answer:
Laws of Exponents. 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.
The value of the given expression is calculated as:
     \[\begin{align}
  & \text{ }{{11}^{8}}\div \left( {{11}^{10}}\div {{11}^{16}} \right) \\
 & ={{11}^{8}}\div \left( {{11}^{10-16}} \right) \\
 & ={{11}^{8}}\div {{11}^{-6}} \\
 & ={{11}^{8-\left( -6 \right)}} \\
 & ={{11}^{8+6}} \\
 & ={{11}^{14}} \\
\end{align}\]
Therefore, value of \[{{11}^{8}}\div \left( {{11}^{10}}\div {{11}^{16}} \right)\] is \[{{11}^{14}}\].

Note: It should be kept in mind that the laws of exponent are applied only when the base is equal. Since 10 and 16 are of different signs, they are subtracted, which gives us 6 and 16 is the larger number which has negative sign, hence a negative sign is assigned to 6. The following rule for multiplication of signs is used:
          \[\left( - \right)\times \left( - \right)=\left( + \right)\]
Therefore, \[8-\left( -6 \right)\] becomes \[8+6\]. We should not get confused while solving this. Simply writing \[8-6\] will give us wrong results. The numbers are subtracted along with their respective signs. If the rules of exponents are not applied, the expression can be solved by expanding. \[{{11}^{10}}\] is expanded as 11 multiplied for 10 times. Similarly, \[{{11}^{16}}\] is expanded as 11 multiplied for 16 times and \[{{11}^{8}}\] is 11 multiplied for 8 times. On expanding, many 11s cancel each other out and only 14 of them are left. This gives us \[{{11}^{14}}\]. However, applying laws of exponent saves us time and is more efficient.