Question

Simplify: $ {\left[ {7{{\left( {{{81}^{\dfrac{1}{4}}} + {{256}^{\dfrac{1}{4}}}} \right)}^{\dfrac{1}{4}}}} \right]^4} $

Answer 1

Hint: The given question is based on the exponent and power. For simplify this problem we first simplify small bracket in which we convert $ 81 $ and $ 256 $ in the power $ 4 $ then we use law of exponent $ {\left( {{a^m}} \right)^n} = {a^{m \times n}} $ . Then we add the resulting number.
After that we again convert this addition into power $ 4 $ and again we use the above property. Then we multiply its result into $ 7 $ and finally we solve power $ 4 $ by multiplying the result itself four times and get the required answer.
In this problem we should know the laws of exponent. By using this law we can simplify this problem in a simple way.

Complete step-by-step answer:
In the given problem we have to simplify $ {\left[ {7{{\left( {{{81}^{\dfrac{1}{4}}} + {{256}^{\dfrac{1}{4}}}} \right)}^{\dfrac{1}{4}}}} \right]^4} $
We first convert $ 81 $ and $ 256 $ in the power $ 4 $ as we know that $ {\left( 3 \right)^4} = 81 $ and $ {\left( 4 \right)^4} = 256 $ then we get
 $ {\left[ {7{{\left( {{{\left( {{3^4}} \right)}^{\dfrac{1}{4}}} + {{\left( {{4^4}} \right)}^{\dfrac{1}{4}}}} \right)}^{\dfrac{1}{4}}}} \right]^4} $
Then we use law of exponent $ {\left( {{a^m}} \right)^n} = {a^{m \times n}} $ we get
 $ \Rightarrow {\left[ {7{{\left( {{3^{4 \times \dfrac{1}{4}}} + {4^{4 \times \dfrac{1}{4}}}} \right)}^{\dfrac{1}{4}}}} \right]^4} $
 $ \Rightarrow {\left[ {7{{\left( {3 + 4} \right)}^{\dfrac{1}{4}}}} \right]^4} $
 $ \Rightarrow {\left[ {7{{\left( 7 \right)}^{\dfrac{1}{4}}}} \right]^4} $
The above expression can be written as
 $ \Rightarrow {\left[ {{7^1} \times {7^{\dfrac{1}{4}}}} \right]^4} $
Now we use the law of exponent $ {a^m} \times {a^n} = {a^{m + n}} $
 $ \Rightarrow {\left[ {{7^{1 + \dfrac{1}{4}}}} \right]^4} $
By taking $ 4 $ as LCM in power of $ 7 $
 $
   \Rightarrow {\left( {{7^{\dfrac{{4 + 1}}{4}}}} \right)^4} \\
   \Rightarrow {\left( {{7^{\dfrac{5}{4}}}} \right)^4} \;
  $
Then we again use the law of exponent $ {\left( {{a^m}} \right)^n} = {a^{m \times n}} $ we get
 $ \Rightarrow {7^{\dfrac{5}{4} \times 4}} $
 $ \Rightarrow {7^5} = 7 \times 7 \times 7 \times 7 \times 7 $
On multiplying $ 7 $ itself $ 5 $ times, we get
 $ \Rightarrow 16807 $
So, the correct answer is “16807”.

Note: It should be noted that the exponent notation is a simple way to express repeated multiplication of the same number.
Very large and very small numbers can be expressed in standard form using exponent notation. To solve the problems based on exponent and power we use laws of exponent.
Some laws of exponent are:
 $
  (i){a^m} \times {a^n} = {a^{m + n}} \\
  (ii){a^m} \div {a^n} = {a^{m - n}} \\
  \left( {iii} \right){\left( {{a^m}} \right)^n} = {a^{m \times n}} \\
  \left( {iv} \right){a^m} \times {b^m} = {\left( {ab} \right)^m} \\
  \left( v \right){a^0} = 1 \\
  \left( {vi} \right)\dfrac{{{a^m}}}{{{b^m}}} = {\left( {\dfrac{a}{b}} \right)^m} \;
  $
By using these laws we can simplify the problems of large and small power in a simple way. So we should remember these laws to simplify the problems based on exponent notation.