Not simplifying the square roots individually

Oct 13, 2025 by windailys.com 108 views

Understanding the Problem

When dealing with mathematical expressions, it's essential to simplify them to make them easier to work with. In this case, we're given the expression $\sqrt{\frac{1}{12}}-\left(\sqrt{-\frac{1}{6}}\right)^2$ . Our goal is to simplify this expression and find its value.

Breaking Down the Expression

To simplify the expression, let's break it down into smaller parts. We can start by simplifying the square roots individually.

Simplifying the First Square Root

The first square root is $\sqrt{\frac{1}{12}}$ . To simplify this, we can find the square root of the numerator and the denominator separately.

$\sqrt{\frac{1}{12}} = \frac{\sqrt{1}}{\sqrt{12}}$

Since the square root of 1 is 1, we can simplify this further.

$\frac{\sqrt{1}}{\sqrt{12}} = \frac{1}{\sqrt{12}}$

Now, let's simplify the denominator. The square root of 12 can be written as $\sqrt{4 \times 3}$ .

$\sqrt{12} = \sqrt{4 \times 3} = \sqrt{4} \times \sqrt{3} = 2\sqrt{3}$

So, the first square root simplifies to:

$\frac{1}{\sqrt{12}} = \frac{1}{2\sqrt{3}}$

Simplifying the Second Square Root

The second square root is $\left(\sqrt{-\frac{1}{6}}\right)^2$ . To simplify this, we need to remember that the square of a square root is the original value.

$\left(\sqrt{-\frac{1}{6}}\right)^2 = -\frac{1}{6}$

Combining the Simplified Square Roots

Now that we've simplified the individual square roots, we can combine them to simplify the original expression.

$\sqrt{\frac{1}{12}}-\left(\sqrt{-\frac{1}{6}}\right)^2 = \frac{1}{2\sqrt{3}} - \left(-\frac{1}{6}\right)$

Rationalizing the Denominator

To simplify the expression further, we can rationalize the denominator. This involves multiplying the numerator and denominator by the conjugate of the denominator.

$\frac{1}{2\sqrt{3}} - \left(-\frac{1}{6}\right) = \frac{1}{2\sqrt{3}} + \frac{1}{6}$

To rationalize the denominator, we can multiply the numerator and denominator by $\sqrt{3}$ .

$\frac{1}{2\sqrt{3}} + \frac{1}{6} = \frac{\sqrt{3}}{6} + \frac{1}{6}$

Simplifying the Expression

Now that we've rationalized the denominator, we can simplify the expression further.

$\frac{\sqrt{3}}{6} + \frac{1}{6} = \frac{\sqrt{3} + 1}{6}$

Conclusion

In conclusion, the simplified expression is $\frac{\sqrt{3} + 1}{6}$ . This is the final value of the original expression.

Final Answer

The final answer is $\boxed{\frac{\sqrt{3} + 1}{6}}$ .

Understanding the Problem

Q&A

Q: What is the first step in simplifying the expression?

A: The first step in simplifying the expression is to break it down into smaller parts. We can start by simplifying the square roots individually.

Q: How do we simplify the first square root?

A: To simplify the first square root, we can find the square root of the numerator and the denominator separately. This gives us $\frac{\sqrt{1}}{\sqrt{12}}$ . Since the square root of 1 is 1, we can simplify this further to $\frac{1}{\sqrt{12}}$ .

Q: How do we simplify the denominator of the first square root?

A: The denominator of the first square root is $\sqrt{12}$ . We can simplify this by writing it as $\sqrt{4 \times 3}$ . This gives us $\sqrt{4} \times \sqrt{3} = 2\sqrt{3}$ .

Q: What is the simplified form of the first square root?

A: The simplified form of the first square root is $\frac{1}{2\sqrt{3}}$ .

Q: How do we simplify the second square root?

A: The second square root is $\left(\sqrt{-\frac{1}{6}}\right)^2$ . To simplify this, we need to remember that the square of a square root is the original value. This gives us $-\frac{1}{6}$ .

Q: How do we combine the simplified square roots?

A: Now that we've simplified the individual square roots, we can combine them to simplify the original expression. This gives us $\frac{1}{2\sqrt{3}} - \left(-\frac{1}{6}\right)$ .

Q: How do we rationalize the denominator?

A: To simplify the expression further, we can rationalize the denominator. This involves multiplying the numerator and denominator by the conjugate of the denominator. This gives us $\frac{\sqrt{3}}{6} + \frac{1}{6}$ .

Q: What is the final simplified form of the expression?

A: The final simplified form of the expression is $\frac{\sqrt{3} + 1}{6}$ .

Common Mistakes

When simplifying the expression, there are a few common mistakes to watch out for.

Not simplifying the square roots individually: It's essential to simplify the square roots individually before combining them.
Not rationalizing the denominator: Rationalizing the denominator is a crucial step in simplifying the expression.
Not combining the simplified square roots: Combining the simplified square roots is the final step in simplifying the expression.

Conclusion

In conclusion, simplifying the expression $\sqrt{\frac{1}{12}}-\left(\sqrt{-\frac{1}{6}}\right)^2$ requires breaking it down into smaller parts, simplifying the square roots individually, and rationalizing the denominator. By following these steps, we can simplify the expression to its final form.

Final Answer

The final answer is $\boxed{\frac{\sqrt{3} + 1}{6}}$ .