Question

Solve the given expression, \[\left( {{2}^{0}}+{{4}^{9}} \right)\div 2\] .

Answer 1

Hint: First of all, replace 4 by \[{{2}^{2}}\] . Now, we know the property that any number having 0 as its exponent is equal to 1 . We also know the property, \[\dfrac{{{x}^{m}}}{{{x}^{n}}}={{x}^{m-n}}\] . Using these two properties, simplify the given equation and solve it further.

Complete step-by-step answer:
According to the question, we have the expression \[\left( {{2}^{0}}+{{4}^{9}} \right)\div 2\] ………………….(1)
In this expression, we have the numbers \[{{2}^{0}}\] , \[{{4}^{9}}\] and 2. Here we have 0 and 9 as the exponents of 2 and 4.
We know that 4 can be written as the square of 2.
\[4={{2}^{2}}\] …………………….(3)
Using equation (3) and transforming equation (1), we get
\[\left( {{2}^{0}}+{{\left( {{2}^{2}} \right)}^{9}} \right)\div 2\] ……………………(4)
We know the formula, \[{{({{x}^{m}})}^{n}}={{x}^{mn}}\] .
Replacing x by 2, m by 2, and n by 9 in the formula, we get
\[{{({{2}^{2}})}^{9}}={{2}^{2.9}}={{2}^{18}}\] ………………………..(5)
Now, we have to replace \[{{({{2}^{2}})}^{9}}\] by \[{{2}^{18}}\] in equation (4).
Using equation (5) and replacing \[{{({{2}^{2}})}^{9}}\] by \[{{2}^{18}}\] in equation (4), we get
\[\left( {{2}^{0}}+{{2}^{18}} \right)\div 2\] ………………….(6)
We know the property that any number having 0 as its exponent is equal to 1. Using this property, we can simplify equation (6).
Now, simplifying equation (6), we get
\[\begin{align}
  & \dfrac{\left( {{2}^{0}}+{{2}^{18}} \right)}{2} \\
 & =\dfrac{1+{{2}^{18}}}{2} \\
\end{align}\]
Now, breaking the above expression as the summation of the number \[\dfrac{1}{2}\] and \[\dfrac{{{2}^{18}}}{2}\] , we get
\[=\dfrac{1+{{2}^{18}}}{2}\]
\[=\dfrac{1}{2}+\dfrac{{{2}^{18}}}{2}\] ………………….(7)
We also know the property, \[\dfrac{{{x}^{m}}}{{{x}^{n}}}={{x}^{m-n}}\] .
Replacing x by 2, m by 18, and n by 1 in the above property we get,
\[\dfrac{{{2}^{18}}}{{{2}^{1}}}={{2}^{18-1}}={{2}^{17}}\] ………………….(8)
Using equation (8) and transforming equation (7), we get
\[\begin{align}
  & =\dfrac{1}{2}+\dfrac{{{2}^{18}}}{2} \\
 & =0.5+{{2}^{18-1}} \\
 & =0.5+{{2}^{17}} \\
\end{align}\]
Hence, the value of \[\left( {{2}^{0}}+{{4}^{9}} \right)\div 2\] is \[(0.5+{{2}^{17}})\] .

Note: In this question, one can make a silly mistake in the property that any number having 0 as its exponent is equal to 0. Using this property one can replace the number \[{{2}^{0}}\] by 0 in the expression \[\left( {{2}^{0}}+{{4}^{9}} \right)\div 2\] . This is wrong. Therefore, we have to keep the correct property in our mind.