Solve: (√20 × √4)/√5 - Simplifying Square Root Fractions

Question

Solve the following exercise:

2045= \frac{\sqrt{20}\cdot\sqrt{4}}{\sqrt{5}}=

Video Solution

Solution Steps

00:00 Solve the following problem
00:03 When multiplying the square root of a number (A) by the square root of another number (B)
00:06 The result equals the square root of their product (A multiplied by B)
00:10 Apply this formula to our exercise and calculate the multiplication
00:24 The square root of a number (A) divided by square root of a number (B)
00:27 Is the same as the square root of the fraction (A divided by B)
00:33 We will apply this formula to our exercise and convert to the square root of a fraction:
00:37 Calculate 80 divided by 5
00:40 Break down 16 into 4 squared
00:43 This is the solution

Step-by-Step Solution

Introduction:

We will address the following three laws of exponents:

a. Definition of root as an exponent:

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

b. The law of exponents for exponents applied to multiplication of terms in parentheses:

(ab)n=anbn (a\cdot b)^n=a^n\cdot b^n

c. The law of exponents for exponents applied to division of terms in parentheses:

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

Note:

(1). By combining the two laws of exponents mentioned in a (in the first and third steps ) and b (in the second step ), we can obtain a new rule:

abn=(ab)1n=a1nb1n=anbnabn=anbn \sqrt[n]{a\cdot b}=\\ (a\cdot b)^{\frac{1}{n}}=\\ a^{\frac{1}{n}}\cdot b^{\frac{1}{n}}=\\ \sqrt[n]{a}\cdot \sqrt[n]{ b}\\ \downarrow\\ \boxed{\sqrt[n]{a\cdot b}=\sqrt[n]{a}\cdot \sqrt[n]{ b}}

Specifically for the fourth root we obtain the following:

ab=ab \boxed{ \sqrt{a\cdot b}=\sqrt{a}\cdot \sqrt{ b}}

(2). Note that by combining the two laws of exponents mentioned in a (in the first and third steps) and c (in the second step), we can obtain another new rule:

abn=(ab)1n=a1nb1n=anbnabn=anbn \sqrt[n]{\frac{a}{b}}=\\ (\frac{a}{b})^{\frac{1}{n}}=\\ \frac{a^{\frac{1}{n}}}{ b^{\frac{1}{n}}}=\\ \frac{\sqrt[n]{a}}{ \sqrt[n]{ b}}\\ \downarrow\\ \boxed{ \sqrt[n]{\frac{a}{b}}=\frac{\sqrt[n]{a}}{ \sqrt[n]{ b}}}

Specifically for the fourth root we obtain the following:

ab=ab \boxed{ \sqrt{\frac{a}{ b}}=\frac{\sqrt{a}}{\sqrt{ b}} }

Therefore, in solving the problem, meaning - in simplifying the given expression, we will use the two new rules that we studied in the introduction:

(1).

ab=ab \boxed{ \sqrt{a\cdot b}=\sqrt{a}\cdot \sqrt{ b}}

(2).

ab=ab \boxed{ \sqrt{\frac{a}{ b}}=\frac{\sqrt{a}}{\sqrt{ b}} }

We'll start by simplifying the expression in the numerator using the rule that we studied in the introduction (1) (however this time in the opposite direction, meaning we'll insert the multiplication of roots as a multiplication of terms under the same root) Then we'll proceed to perform the multiplication under the root in the numerator:

2045=2045=805= \frac{\sqrt{20}\cdot\sqrt{4}}{\sqrt{5}}= \\ \frac{\sqrt{20\cdot4}}{\sqrt{5}}= \\ \frac{\sqrt{80}}{\sqrt{5}}= \\ We'll continue to simplify the fraction, using the rule we received in the introduction (2) (however in the opposite direction, meaning we'll insert the division of roots as a division of terms under the same root) Then we'll proceed to reduce the fraction under the root:

805=805=16=4 \frac{\sqrt{80}}{\sqrt{5}}= \\ \\ \sqrt{\frac{80}{5}}=\\ \sqrt{16}=\\ \boxed{4}

In the final stage, after reducing the fraction under the root, we used the known fourth root of the number 16.

Let's summarize the simplification process of the expression in the problem:

2045=805=16=4 \frac{\sqrt{20}\cdot\sqrt{4}}{\sqrt{5}}= \\ \frac{\sqrt{80}}{\sqrt{5}}= \\ \sqrt{16}=\\ \boxed{4}

Therefore, the correct answer is answer B.

Answer

4 4