Solve: (Square Root of 4 × Square Root of 5) ÷ Square Root of 10 = x

Question

Solve the following equation:

4510=x \frac{\sqrt{4}\cdot\sqrt{5}}{\sqrt{10}}=x

Video Solution

Solution Steps

00:00 Solve the following problem
00:03 When multiplying the root of a number (A) by the root of another number (B)
00:06 The result equals the root of their product (A times B)
00:09 Apply this formula to our exercise and calculate the multiplication
00:21 The root of number (A) divided by root of number (B)
00:24 Is the same as the root of the fraction (A divided by B)
00:27 Apply this formula to our exercise and convert it to the root of a fraction
00:30 Let's calculate 20 divided by 10
00:33 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 an exponent applied to a product in parentheses:

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

c. The law of exponents for an exponent applied to a quotient 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). Similarly, note that by combining the two laws of exponents mentioned in a (in the first and third steps later) and c (in the second step later), 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 apply the two new rules that we received 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 will start by simplifying the expression in the numerator using the rule that we examined in the introduction (1) (however this time in the opposite direction, meaning we will insert the product of roots as a product of terms under the same root) Then we will proceed to perform the multiplication under the root in the numerator:

4510=4510=2010= \frac{\sqrt{4}\cdot\sqrt{5}}{\sqrt{10}}= \\ \frac{\sqrt{4\cdot5}}{\sqrt{10}}= \\ \frac{\sqrt{20}}{\sqrt{10}}= \\ We will then simplify the fraction, using the second rule that we examined in the introduction (2) (once again in the opposite direction, meaning we will insert the quotient of roots as a quotient of terms under the same root) Then we will proceed to reduce the fraction under the root:

2010=2010=2 \frac{\sqrt{20}}{\sqrt{10}}= \\ \sqrt{\frac{20}{10}}=\\ \boxed{\sqrt{2}}

Summarize the process of simplifying the expression in the problem:

4510=2010=2 \frac{\sqrt{4}\cdot\sqrt{5}}{\sqrt{10}}= \\ \frac{\sqrt{20}}{\sqrt{10}}= \\ \boxed{\sqrt{2}}

Therefore, the correct answer is answer c.

Answer

2 \sqrt{2}