Solve the Square Root Expression: Simplifying √(2/4)

Question

Solve the following exercise:

24= \sqrt{\frac{2}{4}}=

Video Solution

Solution Steps

00:00 Solve the following problem
00:04 The root of a fraction (A over B)
00:07 Equals the root of the numerator(A) over the root of the denominator(B)
00:10 Apply this formula to our exercise
00:20 Break down 4 into 2 times 2
00:26 The root of a product (A times B) equals the product of their separate roots
00:32 Apply this formula to our exercise
00:37 Simplify wherever possible
00:40 This is the solution

Step-by-Step Solution

Simplify the following expression:

Begin by reducing the fraction under the square root:

24=12= \sqrt{\frac{2}{4}}= \\ \sqrt{\frac{1}{2}}=

Apply two exponent laws:

A. Definition of root as a power:

an=a1n \sqrt[n]{a}=a^{\frac{1}{n}}

B. The power law for powers applied to terms in parentheses:

(ab)n=anbn \big(\frac{a}{b}\big)^n=\frac{a^n}{b^n}

Let's return to the expression that we obtained. Apply the law mentioned in A and convert the square root to a power:

12=(12)12= \sqrt{\frac{1}{2}}=\\ \big(\frac{1}{2}\big)^{\frac{1}{2}}=

Next use the power law mentioned in B, apply the power separately to the numerator and denominator.

In the next step remember that raising the number 1 to any power will always result in 1.

In the fraction's denominator we'll return to the root notation, again, using the power law mentioned in A (in the opposite direction):

(12)12=112212=12 \big(\frac{1}{2}\big)^{\frac{1}{2}}= \\ \frac{1^{\frac{1}{2}}}{2^{\frac{1}{2}}}=\\ \boxed{\frac{1}{\sqrt{2}}}\\ Let's summarize the simplification of the given expression:

24=12=112212=12 \sqrt{\frac{2}{4}}= \\ \sqrt{\frac{1}{2}}= \\ \frac{1^{\frac{1}{2}}}{2^{\frac{1}{2}}}=\\ \boxed{\frac{1}{\sqrt{2}}}\\ Therefore, the correct answer is answer D.

Answer

12 \frac{1}{\sqrt{2}}