Question

How do you simplify ${\left( {\dfrac{1}{4}} \right)^{ - \dfrac{1}{2}}}$ ?

Answer 1

Hint: To solve this question, we use the rules of exponents to simplify the expression and easily solve it. The two rules of exponents that we use are $a = {a^1}$ and ${\left( {{a^x}} \right)^b} = {a^{xb}}$. We use these two rules to solve and simplify our expression to get our required answer.

Complete step by step solution:
In this question, we are asked to evaluate the expression ${\left( {\dfrac{1}{4}} \right)^{ - \dfrac{1}{2}}}$. Now according to the rules of exponents-
$a = {a^1}$ and ${\left( {{a^x}} \right)^b} = {a^{xb}}$
Thus, we can rewrite the given expression as:
$ \Rightarrow {\left( {\dfrac{{{1^1}}}{{{4^1}}}} \right)^{ - \dfrac{1}{2}}}$
Applying the second rule of exponents, we get:
$ \Rightarrow \left( {\dfrac{{{1^{ - \dfrac{1}{2}}}}}{{{4^{ - \dfrac{1}{2}}}}}} \right)$
If we interchange the numerator and denominator, then the signs are also changed according to the rule $\dfrac{1}{{{a^x}}} = {a^{ - x}}$
Thus, we have $\left( {\dfrac{{{4^{\dfrac{1}{2}}}}}{{{1^{\dfrac{1}{2}}}}}} \right)$
$ \Rightarrow \dfrac{{\sqrt 4 }}{{\sqrt 1 }} = 2$

The value of the given expression is 2.

Note: Exponents consist of two parts – the base and the power. An exponent function tells us how many times we need to multiply the base number to get a certain number. For example ${5^3} = 5 \times 5 \times 5$
Here five is the base while three is the power to which it is raised to. The equation above is said as “ five raised to the power of three”. The power of two can also be said to be “squared” and the power of three can be said to be “cubed”.
Some common properties of exponents are:
Product of powers rule:
When multiplying bases of the same value, their exponents simply get added together. For example ${4^5} \times {4^3} = {4^{\left( {5 + 3} \right)}}$
Quotients of powers rule:
When dividing bases of the same value, their resultant exponent is the difference of the two powers of the bases. For example $\dfrac{{{4^8}}}{{{4^6}}} = {4^{\left( {8 - 6} \right)}}$
Power of a power rule:
When a number raised to a certain power, is then again raised to another power – then both the powers get multiplied. For example ${\left( {{x^5}} \right)^4} = {x^{20}}$