Simplify The Following By Using Surds:$\[ \frac{\sqrt{75}+\sqrt{12}-\sqrt{108}}{\sqrt{147}} \\]

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Introduction

In mathematics, surds are expressions that involve the square root of a number. They are often used in algebra and geometry to simplify complex expressions. In this article, we will simplify the given expression using surds.

The Given Expression

The given expression is:

75+12−108147\frac{\sqrt{75}+\sqrt{12}-\sqrt{108}}{\sqrt{147}}

Simplifying the Expression

To simplify the expression, we need to simplify each term separately. Let's start by simplifying the square roots.

Simplifying the Square Roots

  • 75\sqrt{75} can be simplified as 25×3=53\sqrt{25 \times 3} = 5\sqrt{3}
  • 12\sqrt{12} can be simplified as 4×3=23\sqrt{4 \times 3} = 2\sqrt{3}
  • 108\sqrt{108} can be simplified as 36×3=63\sqrt{36 \times 3} = 6\sqrt{3}
  • 147\sqrt{147} can be simplified as 49×3=73\sqrt{49 \times 3} = 7\sqrt{3}

Substituting the Simplified Square Roots

Now that we have simplified the square roots, let's substitute them back into the original expression.

53+23−6373\frac{5\sqrt{3}+2\sqrt{3}-6\sqrt{3}}{7\sqrt{3}}

Combining Like Terms

Now that we have substituted the simplified square roots, let's combine like terms.

53+23−6373=1373\frac{5\sqrt{3}+2\sqrt{3}-6\sqrt{3}}{7\sqrt{3}} = \frac{1\sqrt{3}}{7\sqrt{3}}

Cancelling Out the Common Factor

Now that we have combined like terms, let's cancel out the common factor.

1373=17\frac{1\sqrt{3}}{7\sqrt{3}} = \frac{1}{7}

Conclusion

In this article, we simplified the given expression using surds. We started by simplifying each term separately, then substituted the simplified square roots back into the original expression. Finally, we combined like terms and cancelled out the common factor to get the final answer.

Final Answer

The final answer is 17\boxed{\frac{1}{7}}.

Importance of Surds

Surds are an important concept in mathematics, particularly in algebra and geometry. They are used to simplify complex expressions and are a fundamental building block for more advanced mathematical concepts.

Real-World Applications

Surds have many real-world applications, including:

  • Engineering: Surds are used in engineering to simplify complex calculations and to design more efficient systems.
  • Physics: Surds are used in physics to describe the behavior of particles and to calculate the energy of systems.
  • Computer Science: Surds are used in computer science to simplify complex algorithms and to optimize system performance.

Common Mistakes

When simplifying expressions using surds, there are several common mistakes to avoid:

  • Not simplifying the square roots: Failing to simplify the square roots can lead to incorrect answers.
  • Not combining like terms: Failing to combine like terms can lead to incorrect answers.
  • Not cancelling out the common factor: Failing to cancel out the common factor can lead to incorrect answers.

Tips and Tricks

Here are some tips and tricks for simplifying expressions using surds:

  • Simplify the square roots first: Simplifying the square roots first can make it easier to simplify the expression.
  • Combine like terms carefully: Combining like terms carefully can help to avoid mistakes.
  • Cancel out the common factor carefully: Canceling out the common factor carefully can help to avoid mistakes.

Conclusion

In conclusion, simplifying expressions using surds is an important concept in mathematics. By simplifying each term separately, substituting the simplified square roots back into the original expression, combining like terms, and cancelling out the common factor, we can simplify complex expressions and get the final answer.

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Q1: What are surds?

A1: Surds are expressions that involve the square root of a number. They are often used in algebra and geometry to simplify complex expressions.

Q2: Why are surds important?

A2: Surds are important because they are used to simplify complex expressions and are a fundamental building block for more advanced mathematical concepts.

Q3: How do I simplify a surd?

A3: To simplify a surd, you need to simplify the square root of the number. You can do this by finding the largest perfect square that divides the number.

Q4: What is the difference between a surd and a rational number?

A4: A surd is an irrational number that cannot be expressed as a finite decimal or fraction, whereas a rational number is a number that can be expressed as a finite decimal or fraction.

Q5: Can I simplify a surd with a negative number?

A5: Yes, you can simplify a surd with a negative number. To do this, you need to simplify the square root of the absolute value of the number.

Q6: How do I simplify a surd with a variable?

A6: To simplify a surd with a variable, you need to simplify the square root of the variable. You can do this by finding the largest perfect square that divides the variable.

Q7: Can I simplify a surd with a fraction?

A7: Yes, you can simplify a surd with a fraction. To do this, you need to simplify the square root of the numerator and the denominator separately.

Q8: How do I simplify a surd with a complex number?

A8: To simplify a surd with a complex number, you need to simplify the square root of the real and imaginary parts separately.

Q9: Can I simplify a surd with a trigonometric function?

A9: Yes, you can simplify a surd with a trigonometric function. To do this, you need to simplify the square root of the trigonometric function.

Q10: How do I check if a surd is simplified?

A10: To check if a surd is simplified, you need to check if the square root of the number is simplified. You can do this by checking if the number is a perfect square.

Q11: Can I use a calculator to simplify a surd?

A11: Yes, you can use a calculator to simplify a surd. However, you need to make sure that the calculator is set to the correct mode and that the surd is entered correctly.

Q12: How do I simplify a surd with a large number?

A12: To simplify a surd with a large number, you need to simplify the square root of the number. You can do this by finding the largest perfect square that divides the number.

Q13: Can I simplify a surd with a decimal number?

A13: Yes, you can simplify a surd with a decimal number. To do this, you need to simplify the square root of the decimal number.

Q14: How do I simplify a surd with a mixed number?

A14: To simplify a surd with a mixed number, you need to simplify the square root of the whole number and the fraction separately.

Q15: Can I simplify a surd with a negative fraction?

A15: Yes, you can simplify a surd with a negative fraction. To do this, you need to simplify the square root of the absolute value of the fraction.

Conclusion

In conclusion, simplifying expressions using surds is an important concept in mathematics. By understanding the basics of surds and how to simplify them, you can simplify complex expressions and get the final answer.