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

Video Solution

Solution Steps

00:10 Let's simplify this problem step by step.

00:13 When you have a square root of a fraction, like A over B,

00:17 you can rewrite it as the square root of A, divided by the square root of B.

00:23 Now, let's apply this rule to our exercise.

00:29 If there's a square root of a multiplication, like A times B,

00:33 you can split it into the square root of A, times the square root of B.

00:38 Let's use this in our exercise as well.

00:45 Break down two hundred twenty-five as fifteen squared.

00:49 Factor X to the power of 4 as X squared, squared.

00:56 Factor nine into three squared.

01:02 Remember, the square root of a square, like A squared, cancels the square.

01:07 Let's apply this to our exercise, and cancel out those squares.

01:24 Break fifteen down to factors of three and five.

01:29 Factor X squared as X times X.

01:34 Simplify wherever you can.

01:37 And there you have it, that's 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: $5x$ .