Simplify: 72 − 4 8 \frac{\sqrt{72}-4}{8} 8 72 ​ − 4 ​

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Introduction

Simplifying mathematical expressions is an essential skill in mathematics, and it requires a deep understanding of various mathematical concepts, including algebra, geometry, and trigonometry. In this article, we will focus on simplifying a specific mathematical expression, $\frac{\sqrt{72}-4}{8}$, using various techniques and strategies.

Understanding the Expression

The given expression is $\frac{\sqrt{72}-4}{8}$. To simplify this expression, we need to understand the properties of square roots and fractions. The square root of a number is a value that, when multiplied by itself, gives the original number. For example, the square root of 16 is 4, because 4 multiplied by 4 equals 16.

Breaking Down the Expression

To simplify the expression, we can start by breaking it down into smaller parts. We can rewrite the expression as $\frac{\sqrt{72}}{8} - \frac{4}{8}$. This allows us to simplify each part separately.

Simplifying the Square Root

The square root of 72 can be simplified by finding the largest perfect square that divides 72. We can rewrite 72 as $36 \times 2$, where 36 is a perfect square. Therefore, $\sqrt{72} = \sqrt{36 \times 2} = 6\sqrt{2}$.

Substituting the Simplified Square Root

Now that we have simplified the square root, we can substitute it back into the expression. We get $\frac{6\sqrt{2}}{8} - \frac{4}{8}$.

Simplifying the Fractions

To simplify the fractions, we can divide both the numerator and the denominator by their greatest common divisor (GCD). In this case, the GCD of 6 and 8 is 2. Therefore, we can simplify the fractions as follows:

$\frac{6\sqrt{2}}{8} = \frac{3\sqrt{2}}{4}$

$\frac{4}{8} = \frac{1}{2}$

Combining the Simplified Fractions

Now that we have simplified the fractions, we can combine them to get the final simplified expression. We get $\frac{3\sqrt{2}}{4} - \frac{1}{2}$.

Rationalizing the Denominator

To rationalize the denominator, we need to multiply the numerator and the denominator by the conjugate of the denominator. In this case, the conjugate of 4 is 4. Therefore, we can multiply the expression by $\frac{4}{4}$ to get:

$\frac{3\sqrt{2}}{4} - \frac{1}{2} = \frac{3\sqrt{2}}{4} - \frac{2}{4}$

Combining the Fractions

Now that we have the same denominator, we can combine the fractions to get the final simplified expression. We get $\frac{3\sqrt{2} - 2}{4}$.

Conclusion

In this article, we simplified the mathematical expression $\frac{\sqrt{72}-4}{8}$ using various techniques and strategies. We broke down the expression into smaller parts, simplified the square root, and combined the simplified fractions to get the final simplified expression. The simplified expression is $\frac{3\sqrt{2} - 2}{4}$.

Final Answer

The final answer is $\boxed{\frac{3\sqrt{2} - 2}{4}}$.

Related Topics

• Simplifying square roots
• Rationalizing denominators
• Combining fractions

Further Reading

• Algebra: A Comprehensive Introduction
• Geometry: A Comprehensive Introduction
• Trigonometry: A Comprehensive Introduction

References

• [1] "Algebra" by Michael Artin
• [2] "Geometry" by Michael Spivak
• [3] "Trigonometry" by I.M. Gelfand
  

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Introduction

In our previous article, we simplified the mathematical expression $\frac{\sqrt{72}-4}{8}$ using various techniques and strategies. In this article, we will answer some frequently asked questions (FAQs) related to the simplification of this expression.

Q&A

Q: What is the square root of 72?

A: The square root of 72 can be simplified by finding the largest perfect square that divides 72. We can rewrite 72 as $36 \times 2$, where 36 is a perfect square. Therefore, $\sqrt{72} = \sqrt{36 \times 2} = 6\sqrt{2}$.

Q: How do I simplify the square root of 72?

A: To simplify the square root of 72, you can find the largest perfect square that divides 72. In this case, we can rewrite 72 as $36 \times 2$, where 36 is a perfect square. Therefore, $\sqrt{72} = \sqrt{36 \times 2} = 6\sqrt{2}$.

Q: What is the greatest common divisor (GCD) of 6 and 8?

A: The greatest common divisor (GCD) of 6 and 8 is 2.

Q: How do I simplify the fractions in the expression?

A: To simplify the fractions, you can divide both the numerator and the denominator by their greatest common divisor (GCD). In this case, the GCD of 6 and 8 is 2. Therefore, we can simplify the fractions as follows:

$\frac{6\sqrt{2}}{8} = \frac{3\sqrt{2}}{4}$

$\frac{4}{8} = \frac{1}{2}$

Q: How do I combine the simplified fractions?

A: To combine the simplified fractions, you can add or subtract the numerators while keeping the same denominator. In this case, we get:

$\frac{3\sqrt{2}}{4} - \frac{1}{2} = \frac{3\sqrt{2} - 2}{4}$

Q: What is the final simplified expression?

A: The final simplified expression is $\frac{3\sqrt{2} - 2}{4}$.

Q: How do I rationalize the denominator?

A: To rationalize the denominator, you need to multiply the numerator and the denominator by the conjugate of the denominator. In this case, the conjugate of 4 is 4. Therefore, we can multiply the expression by $\frac{4}{4}$ to get:

$\frac{3\sqrt{2}}{4} - \frac{1}{2} = \frac{3\sqrt{2}}{4} - \frac{2}{4}$

Q: What is the final answer?

A: The final answer is $\boxed{\frac{3\sqrt{2} - 2}{4}}$.

Conclusion

In this article, we answered some frequently asked questions (FAQs) related to the simplification of the mathematical expression $\frac{\sqrt{72}-4}{8}$. We hope that this article has provided you with a better understanding of the simplification process and has helped you to answer any questions you may have had.

Related Topics

• Simplifying square roots
• Rationalizing denominators
• Combining fractions

Further Reading

• Algebra: A Comprehensive Introduction
• Geometry: A Comprehensive Introduction
• Trigonometry: A Comprehensive Introduction

References

• [1] "Algebra" by Michael Artin
• [2] "Geometry" by Michael Spivak
• [3] "Trigonometry" by I.M. Gelfand