Question

How do you simplify \[{4^{\dfrac{{ - 1}}{2}}}\]?

Answer 1

Hint:We have to simplify the given expression \[{4^{\dfrac{{ - 1}}{2}}}\]. We solve this question using the concept of powers and exponents. We should also have the knowledge about the relation between the negative powers of an exponent and the positive power. We should know how to read a fractional power. First, we will simplify the expression by using the relation of the conversion of the negative power to positive power. Then, we will convert the given number into a fractional number using the power. And then we will split the fraction into its prime factors and then using the power of the exponent we will simplify the given expression.

Complete step by step answer:
Given: simplify \[{4^{\dfrac{{ - 1}}{2}}}\]
Now, we know that the rule of exponents for negative powers is given as:
\[{a^{ - b}} = {\left( {\dfrac{1}{a}} \right)^b}\]
So, using the rule for negative powers of exponents, we can write the expression as:
\[{4^{\dfrac{{ - 1}}{2}}} = {\left( {\dfrac{1}{4}} \right)^{\dfrac{1}{2}}}\]
Now, splitting the above expression, we can write the expression as:
\[{4^{\dfrac{{ - 1}}{2}}} = {\left( {\dfrac{1}{{2 \times 2}}} \right)^{\dfrac{1}{2}}}\]
As, we also know that the property of exponent powers states that we can write the numerator and denominator with power differently.

The expression for the statement is given as:
\[{\left( {\dfrac{1}{a}} \right)^b} = \dfrac{{{1^b}}}{{{a^b}}}\]
So, using the above property we can write the expression as:
\[{4^{\dfrac{{ - 1}}{2}}} = \dfrac{{{{\left( 1 \right)}^{\dfrac{1}{2}}}}}{{{{\left( {2 \times 2} \right)}^{\dfrac{1}{2}}}}}\]
We also know that \[1\] raised to any power is always \[1\].
So, we can write the expression as:
\[{4^{\dfrac{{ - 1}}{2}}} = \dfrac{1}{{{{\left( {2 \times 2} \right)}^{\dfrac{1}{2}}}}}\]
We also know that the fractional power can also be given as:
\[{a^{\dfrac{1}{n}}} = \sqrt[n]{a}\]
Using the above formula, we can write the expression as:
\[{4^{\dfrac{{ - 1}}{2}}} = \dfrac{1}{{\sqrt {2 \times 2} }}\]
On further solving, we get the simplified expression as:
\[{4^{\dfrac{{ - 1}}{2}}} = \dfrac{1}{2}\]

Hence, the simplified expression of \[{4^{\dfrac{{ - 1}}{2}}}\] is \[\dfrac{1}{2}\].

Note:These powers can be positive and negative but can be molded according to our convenience while solving the problem. Also note that cube-root, square-root are fractions with 1 as numerator and respective root in denominator.The various rules and properties for the exponents are as given below :
Product rule : \[{a^n} \times {b^n} = {(a \times b)^n}\]
Quotient rule : \[\dfrac{{{a^n}}}{{{a^m}}} = {a^{n - m}}\]
Zero rule : \[{b^0} = 1\]
One rule : \[{b^1} = b\]