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

Q: Why does √225 equal 15 instead of ±15?

When we write $ \sqrt{225} $, we mean the principal (positive) square root. The symbol √ always gives the non-negative result. If we needed both positive and negative, we'd write ±√225.

Q: How do I know that √(x⁴) = x² and not just x⁴?

Think of it this way: $ x^4 = (x^2)^2 $, so $ \sqrt{x^4} = \sqrt{(x^2)^2} = x^2 $. The square root undoes the squaring, leaving us with $ x^2 $.

Q: Can I simplify the fraction before taking the square root?

Absolutely! You could first simplify $ \frac{225x^4}{9x^2} = 25x^2 $, then take $ \sqrt{25x^2} = 5x $. Both methods give the same answer!

Q: What if x is negative? Does the answer change?

For this problem, we assume $ x > 0 $ to keep things simple. With negative x values, we'd need to use absolute value: $ \sqrt{x^2} = |x| $.

Q: Why do we get 5x instead of 15x as the final answer?

Don't forget the division step! We have $ \frac{15x^2}{3x} $, which simplifies to $ \frac{15}{3} \cdot \frac{x^2}{x} = 5 \cdot x = 5x $.

Radical Simplification with Variable Exponents

Solve the following exercise:

$\sqrt{\frac{225x^4}{9x^2}}=$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

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 written solution

Follow each step carefully to understand the complete solution

Understand the problem

Solve the following exercise:

$\sqrt{\frac{225x^4}{9x^2}}=$

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: $5x$ .

Final Answer

$5x$

Key Points to Remember

Essential concepts to master this topic

Property: Use $\sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}}$ to separate square roots
Technique: Simplify $\sqrt{x^4} = x^2$ since $(x^2)^2 = x^4$
Check: Verify $(5x)^2 = 25x^2$ equals simplified fraction form ✓

Common Mistakes

Avoid these frequent errors

Incorrectly simplifying variable exponents under radicals
Don't assume $\sqrt{x^4} = x^4$ or leave it unsimplified = wrong answer! The radical cancels half the exponent. Always remember $\sqrt{x^n} = x^{n/2}$ when n is even.

Practice Quiz

Test your knowledge with interactive questions

Choose the expression that is equal to the following:

$\sqrt{a}:\sqrt{b}$

FAQ

Everything you need to know about this question

Why does √225 equal 15 instead of ±15?

When we write $\sqrt{225}$ , we mean the principal (positive) square root. The symbol √ always gives the non-negative result. If we needed both positive and negative, we'd write ±√225.

How do I know that √(x⁴) = x² and not just x⁴?

Think of it this way: $x^4 = (x^2)^2$ , so $\sqrt{x^4} = \sqrt{(x^2)^2} = x^2$ . The square root undoes the squaring, leaving us with $x^2$ .

Can I simplify the fraction before taking the square root?

Absolutely! You could first simplify $\frac{225x^4}{9x^2} = 25x^2$ , then take $\sqrt{25x^2} = 5x$ . Both methods give the same answer!

What if x is negative? Does the answer change?

For this problem, we assume $x > 0$ to keep things simple. With negative x values, we'd need to use absolute value: $\sqrt{x^2} = |x|$ .

Why do we get 5x instead of 15x as the final answer?

Don't forget the division step! We have $\frac{15x^2}{3x}$ , which simplifies to $\frac{15}{3} \cdot \frac{x^2}{x} = 5 \cdot x = 5x$ .

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