Square Root and Fraction Simplification with Perfect Squares: 225 and 9

Video Solution

Solution Steps

00:00 Simplify the following problem

00:03 When there's a root of a fraction (A divided by B)

00:06 We can write it as root of the numerator (A) divided by root of the denominator (B)

00:10 Apply this formula to our exercise

00:19 When there's a root of a multiplication (A times B)

00:22 We can divide it into the root of (A) multiplied by the root of (B)

00:25 Apply this formula to our exercise

00:35 Factor 225 into 15 squared

00:39 Factor X to the fourth power to X squared squared

00:46 Factor 9 to 3 squared

00:52 The root of any number (A) squared cancels out the square

00:55 Apply this formula to our exercise, and proceed to cancel out the squares

01:14 Factor 15 into factors of 3 and 5

01:19 Factor X squared into factors of X and X

01:24 Simplify wherever possible

01:27 This is the solution

Step-by-Step Solution

To solve this problem, we'll divide this task into clear steps:

Step 1: Rewrite the expression using the square root property $\sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}}$ . So, $\sqrt{\frac{225x^4}{9x^2}} = \frac{\sqrt{225x^4}}{\sqrt{9x^2}}$ .
Step 2: Simplify $\sqrt{225x^4}$ . This can be broken down as $\sqrt{225} \times \sqrt{x^4}$ .
Step 3: Simplify each part: $\sqrt{225} = 15$ and $\sqrt{x^4} = x^2$ since $(x^2)^2 = x^4$ .
Step 4: Now, simplify $\sqrt{9x^2}$ . This is $\sqrt{9} \times \sqrt{x^2}$ .
Step 5: Simplify these parts: $\sqrt{9} = 3$ and $\sqrt{x^2} = x$ (assuming $x > 0$ for simplification purposes).
Step 6: Putting it all together, $\frac{\sqrt{225x^4}}{\sqrt{9x^2}} = \frac{15x^2}{3x}$ .
Step 7: Simplify the fraction: $\frac{15x^2}{3x} = 5x$ (since $\frac{x^2}{x} = x$ ).

Therefore, the solution to the problem is $5x$ .

In the choices provided, the correct answer that matches our solution is choice 2: $5x$ .