Question

Simplify $\sqrt {4{x^4}} $?

Answer 1

Hint: Solve by transform the number into its prime factors, and then we will get the expression in the exponents then apply the identity ${a^m} \times {b^m} = {\left( {ab} \right)^m}$ then again apply the identity ${\left( {{a^m}} \right)^n} = {a^{m \times n}}$, and then simplify the expression to get the required result.

Complete step by step answer:
Exponents are defined as when an expression or a statement of specific natural numbers are represented as a repeated power by multiplication of its units then the resulting number is called as an exponent. The resulting set of numbers are the same as the original sequence.
Given expression $\sqrt {4{x^4}} $,
This can be rewritten as, ${\left( {4{x^4}} \right)^{\frac{1}{2}}}$,
Now firstly write in prime factors, we get,
$ \Rightarrow 4 = 2 \times 2$,
Now here we can see that 64 can be written as 2 is multiplied 2 times, this can be written as,$ \Rightarrow 4 = 2 \times 2 = {2^2}$,
Therefore the given expression can be written as,
$ \Rightarrow {\left( {4{x^4}} \right)^{\frac{1}{2}}} = {\left( {{2^2} \cdot {x^4}} \right)^{\frac{1}{2}}}$,
Now using exponential identity ${a^m} \times {b^m} = {\left( {ab} \right)^m}$, we get,
$ \Rightarrow {\left( {4{x^4}} \right)^{\frac{1}{2}}} = {\left( {{2^2}} \right)^{\frac{1}{2}}} \cdot {\left( {{x^4}} \right)^{\frac{1}{2}}}$
Now using the exponent identity,${\left( {{a^m}} \right)^n} = {a^{m \times n}}$,
So here we have for the first term , $a = 2$, $m = 2$ and $n = \dfrac{1}{2}$, and for the second term we have, $a = x$, $m = 4$ and $n = \dfrac{1}{2}$,
Now substituting then values in the identity we get,
$ \Rightarrow {\left( {4{x^4}} \right)^{\frac{1}{2}}} = \left( {{2^{2 \times \frac{1}{2}}}} \right) \cdot \left( {{x^{4 \times \frac{1}{2}}}} \right)$,
Now simplifying we get,
$ \Rightarrow {\left( {4{x^4}} \right)^{\frac{1}{2}}} = \left( {{2^1}} \right) \cdot \left( {{x^2}} \right)$,
Now multiplying the powers we get,
$ \Rightarrow {\left( {4{x^4}} \right)^{\frac{1}{2}}} = 2{x^2}$,
So the given expression is equal to $2{x^2}$.
$\therefore $ The simplified form of $\sqrt {4{x^4}} $will be equal to $2{x^2}$.

Note: There are various laws of exponents we should remember and practise in order to solve and understand the exponential concept. The following are some of the exponent laws:
${a^0} = 1$,
${a^m} \times {a^n} = {a^{m + n}}$,
$\dfrac{{{a^m}}}{{{a^n}}} = {a^{m - n}}$,
$\dfrac{1}{{{a^m}}} = {a^{ - m}}$,
${a^m} \times {b^m} = {\left( {ab} \right)^m}$,
$\dfrac{{{a^m}}}{{{b^m}}} = {\left( {\dfrac{a}{b}} \right)^m}$.