Simplify The Expression:${ \frac{\sqrt{27 M^6} - \sqrt{48 M^6}}{\sqrt{12 M^6}} }$

Introduction

When dealing with expressions involving square roots, it's essential to simplify them to make calculations easier and more manageable. In this article, we will focus on simplifying the given expression: $\frac{\sqrt{27 m^6} - \sqrt{48 m^6}}{\sqrt{12 m^6}}$. We will break down the steps involved in rationalizing the denominator and provide a clear explanation of each process.

Understanding the Expression

The given expression involves square roots, which can be simplified by factoring out the perfect squares. The expression is $\frac{\sqrt{27 m^6} - \sqrt{48 m^6}}{\sqrt{12 m^6}}$. To simplify this expression, we need to start by simplifying the square roots in the numerator and denominator.

Simplifying the Square Roots

The square roots in the expression can be simplified by factoring out the perfect squares. The square root of a number can be expressed as the product of the square root of the perfect square and the square root of the remaining factor.

Simplifying the Numerator

The numerator of the expression is $\sqrt{27 m^6} - \sqrt{48 m^6}$. We can simplify the square roots by factoring out the perfect squares.

$\sqrt{27 m^6} = \sqrt{9 m^6 \cdot 3} = 3 m^3 \sqrt{3}$

$\sqrt{48 m^6} = \sqrt{16 m^6 \cdot 3} = 4 m^3 \sqrt{3}$

Substituting these simplified expressions back into the original expression, we get:

$\frac{3 m^3 \sqrt{3} - 4 m^3 \sqrt{3}}{\sqrt{12 m^6}}$

Simplifying the Denominator

The denominator of the expression is $\sqrt{12 m^6}$. We can simplify the square root by factoring out the perfect square.

$\sqrt{12 m^6} = \sqrt{4 m^6 \cdot 3} = 2 m^3 \sqrt{3}$

Substituting this simplified expression back into the original expression, we get:

$\frac{3 m^3 \sqrt{3} - 4 m^3 \sqrt{3}}{2 m^3 \sqrt{3}}$

Rationalizing the Denominator

Now that we have simplified the numerator and denominator, we can rationalize the denominator by multiplying both the numerator and denominator by the conjugate of the denominator.

The conjugate of the denominator is $2 m^3 \sqrt{3}$.

Multiplying both the numerator and denominator by the conjugate, we get:

$\frac{(3 m^3 \sqrt{3} - 4 m^3 \sqrt{3}) \cdot (2 m^3 \sqrt{3})}{(2 m^3 \sqrt{3}) \cdot (2 m^3 \sqrt{3})}$

Expanding the numerator and denominator, we get:

$\frac{6 m^6 \cdot 3 - 8 m^6 \cdot 3}{4 m^6 \cdot 3}$

Simplifying the numerator and denominator, we get:

$\frac{-2 m^6 \cdot 3}{4 m^6 \cdot 3}$

Final Simplification

Now that we have rationalized the denominator, we can simplify the expression further by canceling out any common factors.

The numerator and denominator both have a factor of $3$, which can be canceled out.

$\frac{-2 m^6 \cdot 3}{4 m^6 \cdot 3} = \frac{-2 m^6}{4 m^6}$

The numerator and denominator both have a factor of $m^6$, which can be canceled out.

$\frac{-2 m^6}{4 m^6} = \frac{-2}{4}$

Simplifying the fraction, we get:

$\frac{-2}{4} = -\frac{1}{2}$

Conclusion

In this article, we simplified the given expression: $\frac{\sqrt{27 m^6} - \sqrt{48 m^6}}{\sqrt{12 m^6}}$. We broke down the steps involved in rationalizing the denominator and provided a clear explanation of each process. By simplifying the square roots and rationalizing the denominator, we were able to simplify the expression to its final form: $-\frac{1}{2}$.

Frequently Asked Questions

• Q: What is the process of rationalizing the denominator? A: Rationalizing the denominator involves multiplying both the numerator and denominator by the conjugate of the denominator to eliminate any square roots in the denominator.
• Q: Why is it necessary to simplify the square roots in the expression? A: Simplifying the square roots makes it easier to rationalize the denominator and simplify the expression.
• Q: What is the final simplified form of the expression? A: The final simplified form of the expression is $-\frac{1}{2}$.

Additional Resources

• For more information on simplifying expressions involving square roots, see the article "Simplifying Expressions Involving Square Roots".
• For more information on rationalizing the denominator, see the article "Rationalizing the Denominator: A Step-by-Step Guide".
• For more information on simplifying fractions, see the article "Simplifying Fractions: A Step-by-Step Guide".
  

Introduction

In our previous article, we simplified the expression: $\frac{\sqrt{27 m^6} - \sqrt{48 m^6}}{\sqrt{12 m^6}}$. We broke down the steps involved in rationalizing the denominator and provided a clear explanation of each process. In this article, we will answer some of the most frequently asked questions about simplifying expressions involving square roots and rationalizing the denominator.

Q&A

Q: What is the process of rationalizing the denominator?

A: Rationalizing the denominator involves multiplying both the numerator and denominator by the conjugate of the denominator to eliminate any square roots in the denominator.

Q: Why is it necessary to simplify the square roots in the expression?

A: Simplifying the square roots makes it easier to rationalize the denominator and simplify the expression. By simplifying the square roots, we can eliminate any common factors and make the expression easier to work with.

Q: What is the difference between rationalizing the denominator and simplifying the expression?

A: Rationalizing the denominator involves multiplying both the numerator and denominator by the conjugate of the denominator to eliminate any square roots in the denominator. Simplifying the expression involves reducing the expression to its simplest form by eliminating any common factors.

Q: Can I simplify the expression without rationalizing the denominator?

A: Yes, you can simplify the expression without rationalizing the denominator. However, rationalizing the denominator can make the expression easier to work with and can help you to eliminate any square roots in the denominator.

Q: How do I know when to rationalize the denominator?

A: You should rationalize the denominator when the expression has a square root in the denominator. Rationalizing the denominator can help you to eliminate any square roots in the denominator and make the expression easier to work with.

Q: Can I use a calculator to simplify the expression?

A: Yes, you can use a calculator to simplify the expression. However, it's always a good idea to check your work by hand to make sure that the expression is simplified correctly.

Q: What are some common mistakes to avoid when simplifying expressions involving square roots?

A: Some common mistakes to avoid when simplifying expressions involving square roots include:

• Not simplifying the square roots before rationalizing the denominator
• Not multiplying both the numerator and denominator by the conjugate of the denominator
• Not eliminating any common factors in the numerator and denominator
• Not checking your work by hand to make sure that the expression is simplified correctly

Additional Resources

• For more information on simplifying expressions involving square roots, see the article "Simplifying Expressions Involving Square Roots".
• For more information on rationalizing the denominator, see the article "Rationalizing the Denominator: A Step-by-Step Guide".
• For more information on simplifying fractions, see the article "Simplifying Fractions: A Step-by-Step Guide".

Conclusion

In this article, we answered some of the most frequently asked questions about simplifying expressions involving square roots and rationalizing the denominator. We provided clear explanations and examples to help you understand the process of rationalizing the denominator and simplifying expressions involving square roots. By following the steps outlined in this article, you can simplify expressions involving square roots and rationalize the denominator with confidence.

Frequently Asked Questions

• Q: What is the process of rationalizing the denominator? A: Rationalizing the denominator involves multiplying both the numerator and denominator by the conjugate of the denominator to eliminate any square roots in the denominator.
• Q: Why is it necessary to simplify the square roots in the expression? A: Simplifying the square roots makes it easier to rationalize the denominator and simplify the expression.
• Q: What is the difference between rationalizing the denominator and simplifying the expression? A: Rationalizing the denominator involves multiplying both the numerator and denominator by the conjugate of the denominator to eliminate any square roots in the denominator. Simplifying the expression involves reducing the expression to its simplest form by eliminating any common factors.

Related Articles

• Simplifying Expressions Involving Square Roots
• Rationalizing the Denominator: A Step-by-Step Guide
• Simplifying Fractions: A Step-by-Step Guide