Solve the Square Root Expression: Simplifying √(25x^4)

Q: Why do I split √25x⁴ into two separate square roots?

The product property of square roots lets you separate terms: $ \sqrt{25x^4} = \sqrt{25} \cdot \sqrt{x^4} $. This makes each part easier to simplify!

Q: How do I know that √25 equals 5?

Think: what number times itself gives 25? Since $ 5 \times 5 = 25 $, we know $ \sqrt{25} = 5 $.

Q: Why does √x⁴ become x² instead of staying x⁴?

The square root undoes squaring. Since $ x^4 = (x^2)^2 $, taking the square root gives us $ x^2 $. Remember: $ \sqrt{x^n} = x^{n/2} $!

Q: What if x could be negative?

For this problem, we assume x represents a positive value. In advanced algebra, you'd need to consider absolute values, but for now, focus on the basic simplification rules.

Q: How can I check if 5x² is really correct?

Square your answer! $ (5x^2)^2 = 5^2 \cdot (x^2)^2 = 25 \cdot x^4 = 25x^4 $ ✓ This matches the original expression under the square root.

Square Root Properties with Perfect Powers

Solve the following exercise:

$\sqrt{25x^4}=$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Simplify the following expression

00:03 The square root of a number (M) multiplied by the square root of another number (N)

00:07 Equals the square root of their product (M times N)

00:14 Apply this formula to our exercise, and convert from root 1 to two

00:20 Break down 25 to 5 squared

00:24 Break down X to the fourth power into X squared squared

00:30 The square root of any number(M) squared cancels out the square

00:37 Apply this formula to our exercise

00:40 This is the solution

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Solve the following exercise:

$\sqrt{25x^4}=$

Step-by-step solution

To solve the expression $\sqrt{25x^4}$ , we will use the product property of square roots.

First, separate the expression inside the square root into two parts:
$\sqrt{25x^4} = \sqrt{25} \cdot \sqrt{x^4}$ .

Next, simplify each square root:

$\sqrt{25} = 5$ because $5^2 = 25$ .
$\sqrt{x^4} = x^{4/2} = x^2$ since the square root of $x^n$ is $x^{n/2}$ for even $n$ .

Now combine the simplified terms:
$5 \cdot x^2$ .

Therefore, the simplified expression is $5x^2$ .

The correct answer is therefore $\mathbf{5x^2}$ , which corresponds to choice $2$ .

Final Answer

$5x^2$

Key Points to Remember

Essential concepts to master this topic

Product Rule: Split square roots: $\sqrt{ab} = \sqrt{a} \cdot \sqrt{b}$
Power Rule: For even exponents: $\sqrt{x^4} = x^{4/2} = x^2$
Check: Verify $(5x^2)^2 = 25x^4$ matches original expression ✓

Common Mistakes

Avoid these frequent errors

Dividing the exponent by the wrong number
Don't divide x^4 by 2 to get 2x^2 = wrong answer! This confuses coefficient with exponent. Always divide the exponent by 2: x^4 becomes x^(4÷2) = x^2, then multiply by √25 = 5.

Practice Quiz

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Choose the largest value

FAQ

Everything you need to know about this question

Why do I split √25x⁴ into two separate square roots?

The product property of square roots lets you separate terms: $\sqrt{25x^4} = \sqrt{25} \cdot \sqrt{x^4}$ . This makes each part easier to simplify!

How do I know that √25 equals 5?

Think: what number times itself gives 25? Since $5 \times 5 = 25$ , we know $\sqrt{25} = 5$ .

Why does √x⁴ become x² instead of staying x⁴?

The square root undoes squaring. Since $x^4 = (x^2)^2$ , taking the square root gives us $x^2$ . Remember: $\sqrt{x^n} = x^{n/2}$ !

What if x could be negative?

For this problem, we assume x represents a positive value. In advanced algebra, you'd need to consider absolute values, but for now, focus on the basic simplification rules.

How can I check if 5x² is really correct?

Square your answer! $(5x^2)^2 = 5^2 \cdot (x^2)^2 = 25 \cdot x^4 = 25x^4$ ✓ This matches the original expression under the square root.

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Solve the Square Root Expression: Simplifying √(25x^4)

Square Root Properties with Perfect Powers

❤️ Continue Your Math Journey!

Step-by-step video solution

Step-by-step written solution

Understand the problem

Step-by-step solution

Final Answer

Key Points to Remember

Common Mistakes

Practice Quiz

FAQ

Why do I split √25x⁴ into two separate square roots?

How do I know that √25 equals 5?

Why does √x⁴ become x² instead of staying x⁴?

What if x could be negative?

How can I check if 5x² is really correct?

🌟 Unlock Your Math Potential

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