Solve the Square Root Expression: Simplifying √x²

Q: Why doesn't the square root just cancel out the x²?

The square root doesn't cancel - it transforms! Think of $ \sqrt{x^2} $ as asking "what number times itself gives x²?" The answer is x, not x².

Q: What's the difference between √(x²) and (√x)²?

$ \sqrt{x^2} = x $ but $ (\sqrt{x})^2 = x $. They both equal x, but the first simplifies the exponent inside the radical, while the second cancels the square root and square.

Q: Do I always convert square roots to fractional exponents?

It's not required, but it's very helpful! Converting $ \sqrt{x^2} $ to $ (x^2)^{\frac{1}{2}} $ lets you use the power rule clearly and avoid confusion.

Q: What if x is negative? Is √(x²) still equal to x?

Actually, $ \sqrt{x^2} = |x| $ (absolute value of x). For this problem, we assume x is positive, but technically the complete answer includes absolute value signs.

Q: How can I remember the power rule (aᵐ)ⁿ = aᵐⁿ?

Think "multiply the powers"! When you have a power raised to another power, you multiply the exponents together. $ (x^2)^{\frac{1}{2}} = x^{2 \times \frac{1}{2}} = x^1 = x $.

Square Root Simplification with Perfect Square Expressions

Solve the following exercise:

$\sqrt{x^2}=$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:05 Let's simplify the expression together.

00:10 Square root of a number squared cancel each other out.

00:15 And this works when A is zero or more.

00:19 We'll apply this formula in our exercise.

00:22 And that's how we find the solution to the question!

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Solve the following exercise:

$\sqrt{x^2}=$

Step-by-step solution

In order to simplify the given expression, we will use two laws of exponents:

a. The definition of root as an exponent:

$\sqrt[n]{a}=a^{\frac{1}{n}}$ b. The law of exponents for power of a power:

$(a^m)^n=a^{m\cdot n}$

Let's start with converting the square root to an exponent using the law mentioned in a':

$\sqrt{x^2}= \\ \downarrow\\ (x^2)^{\frac{1}{2}}=$ We'll continue using the law of exponents mentioned in b' and perform the exponent operation on the term in parentheses:

$(x^2)^{\frac{1}{2}}= \\ x^{2\cdot\frac{1}{2}}\\ x^1=\\ \boxed{x}$ Therefore, the correct answer is answer a'.

Final Answer

$x$

Key Points to Remember

Essential concepts to master this topic

Definition: Convert square root to fractional exponent: $\sqrt{x^2} = (x^2)^{\frac{1}{2}}$
Power Rule: Multiply exponents: $(x^2)^{\frac{1}{2}} = x^{2 \cdot \frac{1}{2}} = x^1$
Verify: Check that $(\sqrt{x^2})^2 = x^2$ gives back original expression ✓

Common Mistakes

Avoid these frequent errors

Thinking the square root cancels to leave x²
Don't assume $\sqrt{x^2} = x^2$ = wrong answer! The square root and the square power don't just disappear - they interact through exponent rules. Always convert the square root to $(x^2)^{\frac{1}{2}}$ and multiply the exponents.

Practice Quiz

Test your knowledge with interactive questions

$\sqrt{100}=$

FAQ

Everything you need to know about this question

Why doesn't the square root just cancel out the x²?

The square root doesn't cancel - it transforms! Think of $\sqrt{x^2}$ as asking "what number times itself gives x²?" The answer is x, not x².

What's the difference between √(x²) and (√x)²?

$\sqrt{x^2} = x$ but $(\sqrt{x})^2 = x$ . They both equal x, but the first simplifies the exponent inside the radical, while the second cancels the square root and square.

Do I always convert square roots to fractional exponents?

It's not required, but it's very helpful! Converting $\sqrt{x^2}$ to $(x^2)^{\frac{1}{2}}$ lets you use the power rule clearly and avoid confusion.

What if x is negative? Is √(x²) still equal to x?

Actually, $\sqrt{x^2} = |x|$ (absolute value of x). For this problem, we assume x is positive, but technically the complete answer includes absolute value signs.

How can I remember the power rule (aᵐ)ⁿ = aᵐⁿ?

Think "multiply the powers"! When you have a power raised to another power, you multiply the exponents together. $(x^2)^{\frac{1}{2}} = x^{2 \times \frac{1}{2}} = x^1 = x$ .

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Solve the Square Root Expression: Simplifying √x²

Square Root Simplification with Perfect Square Expressions

❤️ 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 doesn't the square root just cancel out the x²?

What's the difference between √(x²) and (√x)²?

Do I always convert square roots to fractional exponents?

What if x is negative? Is √(x²) still equal to x?

How can I remember the power rule (aᵐ)ⁿ = aᵐⁿ?

🌟 Unlock Your Math Potential

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