Question

Find the value of \[2\left[ {\left( {{2^{ - 1}} \times {4^{ - 1}}} \right) \div {2^{ - 2}}} \right]\].

Answer 1

Hint:
Here, we will first simplify the given arithmetic expression by using the Negative Exponent rule. Then we will use the BODMAS rule and simplify further to get the required answer. Arithmetic Expression is defined as an expression with numbers and arithmetic operators.

Formula Used:
Negative Exponent Rule: \[{a^{ - n}} = \dfrac{1}{{{a^n}}}\]

Complete step by step solution:
We are given with an Arithmetic Expression \[2\left[ {\left( {{2^{ - 1}} \times {4^{ - 1}}} \right) \div {2^{ - 2}}} \right]\].
Now, by using the Negative Exponent Rule \[{a^{ - n}} = \dfrac{1}{{{a^n}}}\], we get
\[ \Rightarrow 2\left[ {\left( {{2^{ - 1}} \times {4^{ - 1}}} \right) \div {2^{ - 2}}} \right] = 2\left[ {\left( {\dfrac{1}{2} \times \dfrac{1}{4}} \right) \div \dfrac{1}{{{2^2}}}} \right]\]
We will simplify the arithmetic expression by using the BODMAS rule.
Now, by using the BODMAS Rule, we get
\[ \Rightarrow 2\left[ {\left( {{2^{ - 1}} \times {4^{ - 1}}} \right) \div {2^{ - 2}}} \right] = 2\left[ {\left( {\dfrac{1}{8}} \right) \div \dfrac{1}{{{2^2}}}} \right]\]
Applying the exponent on the terms, we get
\[ \Rightarrow 2\left[ {\left( {{2^{ - 1}} \times {4^{ - 1}}} \right) \div {2^{ - 2}}} \right] = 2\left[ {\left( {\dfrac{1}{8}} \right) \div \dfrac{1}{4}} \right]\]
Now, by dividing the numbers inside the bracket, we get
\[ \Rightarrow 2\left[ {\left( {{2^{ - 1}} \times {4^{ - 1}}} \right) \div {2^{ - 2}}} \right] = 2\left[ {\dfrac{4}{8}} \right]\]
\[ \Rightarrow 2\left[ {\left( {{2^{ - 1}} \times {4^{ - 1}}} \right) \div {2^{ - 2}}} \right] = 2\left[ {\dfrac{1}{2}} \right]\]
Multiplying the numbers and canceling out the numbers, we get
\[ \Rightarrow 2\left[ {\left( {{2^{ - 1}} \times {4^{ - 1}}} \right) \div {2^{ - 2}}} \right] = 1\]

Therefore, the value of \[2\left[ {\left( {{2^{ - 1}} \times {4^{ - 1}}} \right) \div {2^{ - 2}}} \right]\] is \[1\] .

Note:
We know that the BODMAS rule states that the first operation has to be done which is in the brackets, next the operation applies on the indices or order, then it moves on to the division and multiplication and then using addition and subtraction we will simplify the expression. If addition or subtraction and division or multiplication are in the same calculations, then it has to be done from left to right. An arithmetic expression is defined as an expression with the numbers and the arithmetic operators like plus, minus etc. Negative Exponent rule, which says that the negative exponent in the numerator gets changed to the denominator, then the exponent becomes positive.