Apply Surd Rules To Simplify The Following Without The Use Of A Calculator And Give The Answer With A Rational Denominator. Show All Steps:${ \frac{\sqrt{10} - \sqrt{5}}{\sqrt{10}} }$

Introduction

Simplifying surds is an essential skill in mathematics, particularly in algebra and geometry. Surds are expressions that involve the square root of a number, and they can be simplified using various techniques. In this article, we will apply surd rules to simplify the expression $\frac{\sqrt{10} - \sqrt{5}}{\sqrt{10}}$ without the use of a calculator and provide the answer with a rational denominator.

Understanding Surds

Before we dive into the simplification process, let's briefly review what surds are. A surd is an expression that involves the square root of a number, such as $\sqrt{10}$ or $\sqrt{5}$. Surds can be rational or irrational, depending on whether the number under the square root sign is a perfect square or not.

Simplifying the Expression

To simplify the expression $\frac{\sqrt{10} - \sqrt{5}}{\sqrt{10}}$, we will use the following steps:

Step 1: Multiply the Numerator and Denominator by the Conjugate of the Denominator

The conjugate of the denominator $\sqrt{10}$ is also $\sqrt{10}$. To simplify the expression, we will multiply the numerator and denominator by the conjugate of the denominator.

$\frac{\sqrt{10} - \sqrt{5}}{\sqrt{10}} \cdot \frac{\sqrt{10}}{\sqrt{10}}$

Step 2: Simplify the Expression

Now, let's simplify the expression by multiplying the numerator and denominator.

$\frac{(\sqrt{10} - \sqrt{5}) \cdot \sqrt{10}}{\sqrt{10} \cdot \sqrt{10}}$

Step 3: Expand the Numerator

Next, let's expand the numerator by multiplying the two terms.

$\frac{\sqrt{100} - \sqrt{50}}{10}$

Step 4: Simplify the Square Roots

Now, let's simplify the square roots in the numerator.

$\frac{10 - \sqrt{50}}{10}$

Step 5: Simplify the Square Root of 50

The square root of 50 can be simplified as follows:

$\sqrt{50} = \sqrt{25 \cdot 2} = 5\sqrt{2}$

Step 6: Substitute the Simplified Square Root

Now, let's substitute the simplified square root back into the expression.

$\frac{10 - 5\sqrt{2}}{10}$

Step 7: Simplify the Expression

Finally, let's simplify the expression by dividing both the numerator and denominator by 10.

$\frac{10}{10} - \frac{5\sqrt{2}}{10}$

Step 8: Simplify the Expression

Now, let's simplify the expression by evaluating the fractions.

$1 - \frac{\sqrt{2}}{2}$

Step 9: Rationalize the Denominator

To rationalize the denominator, we will multiply the numerator and denominator by 2.

$2 - \sqrt{2}$

Step 10: Simplify the Expression

Finally, let's simplify the expression by combining the terms.

$2 - \sqrt{2}$

Conclusion

In this article, we applied surd rules to simplify the expression $\frac{\sqrt{10} - \sqrt{5}}{\sqrt{10}}$ without the use of a calculator and provided the answer with a rational denominator. We used various techniques, including multiplying the numerator and denominator by the conjugate of the denominator, expanding the numerator, simplifying the square roots, and rationalizing the denominator. The final simplified expression is $2 - \sqrt{2}$.

Key Takeaways

• Surds are expressions that involve the square root of a number.
• To simplify surds, we can use various techniques, including multiplying the numerator and denominator by the conjugate of the denominator, expanding the numerator, simplifying the square roots, and rationalizing the denominator.
• The final simplified expression is $2 - \sqrt{2}$.

Further Reading

If you want to learn more about surds and how to simplify them, here are some additional resources:

• Khan Academy: Surds
• Mathway: Simplifying Surds
• Wolfram Alpha: Surds

References

• "Algebra and Trigonometry" by Michael Sullivan
• "Mathematics for the Nonmathematician" by Morris Kline
• "The Art of Mathematics" by Bela Bollobas
  Simplifying Surds: A Q&A Guide =====================================

Introduction

In our previous article, we explored the concept of surds and how to simplify them using various techniques. In this article, we will answer some frequently asked questions about simplifying surds.

Q: What is a surd?

A: A surd is an expression that involves the square root of a number, such as $\sqrt{10}$ or $\sqrt{5}$. Surds can be rational or irrational, depending on whether the number under the square root sign is a perfect square or not.

Q: How do I simplify a surd?

A: To simplify a surd, you can use various techniques, including:

• Multiplying the numerator and denominator by the conjugate of the denominator
• Expanding the numerator
• Simplifying the square roots
• Rationalizing the denominator

Q: What is the conjugate of a denominator?

A: The conjugate of a denominator is the same expression with the opposite sign. For example, the conjugate of $\sqrt{10}$ is also $\sqrt{10}$.

Q: How do I multiply the numerator and denominator by the conjugate of the denominator?

A: To multiply the numerator and denominator by the conjugate of the denominator, you simply multiply the two expressions together.

Q: What is the difference between a rational and irrational surd?

A: A rational surd is an expression that can be simplified to a rational number, such as $\sqrt{4}$, which simplifies to $2$. An irrational surd is an expression that cannot be simplified to a rational number, such as $\sqrt{2}$.

Q: Can I simplify a surd with a negative number under the square root sign?

A: Yes, you can simplify a surd with a negative number under the square root sign. To do this, you will need to use the imaginary unit, $i$, which is defined as the square root of $-1$.

Q: How do I simplify a surd with a negative number under the square root sign?

A: To simplify a surd with a negative number under the square root sign, you will need to use the following steps:

1. Multiply the numerator and denominator by the conjugate of the denominator.
2. Simplify the square roots.
3. Rationalize the denominator.

Q: Can I simplify a surd with a variable under the square root sign?

A: Yes, you can simplify a surd with a variable under the square root sign. To do this, you will need to use the following steps:

1. Multiply the numerator and denominator by the conjugate of the denominator.
2. Simplify the square roots.
3. Rationalize the denominator.

Q: How do I simplify a surd with a variable under the square root sign?

A: To simplify a surd with a variable under the square root sign, you will need to use the following steps:

1. Multiply the numerator and denominator by the conjugate of the denominator.
2. Simplify the square roots.
3. Rationalize the denominator.

Conclusion

In this article, we answered some frequently asked questions about simplifying surds. We covered topics such as what a surd is, how to simplify a surd, and how to simplify a surd with a negative number under the square root sign. We also covered how to simplify a surd with a variable under the square root sign.

Key Takeaways

• Surds are expressions that involve the square root of a number.
• To simplify a surd, you can use various techniques, including multiplying the numerator and denominator by the conjugate of the denominator, expanding the numerator, simplifying the square roots, and rationalizing the denominator.
• The conjugate of a denominator is the same expression with the opposite sign.
• A rational surd is an expression that can be simplified to a rational number, while an irrational surd is an expression that cannot be simplified to a rational number.

Further Reading

If you want to learn more about surds and how to simplify them, here are some additional resources:

• Khan Academy: Surds
• Mathway: Simplifying Surds
• Wolfram Alpha: Surds

References

• "Algebra and Trigonometry" by Michael Sullivan
• "Mathematics for the Nonmathematician" by Morris Kline
• "The Art of Mathematics" by Bela Bollobas