Express The Following Expression In Terms Of Simplified Surds:${ \frac{1}{\sqrt{75}+\sqrt{18}-\sqrt{27}} }$
Introduction
Surds are mathematical expressions that involve the square root of a number. They are often used in algebra and geometry to represent irrational numbers. In this article, we will explore how to simplify surds and express the given expression in terms of simplified surds.
Understanding Surds
A surd is a mathematical expression that involves the square root of a number. It is often represented as √a, where a is a number. For example, √2, √3, and √5 are all surds. Surds can be added, subtracted, multiplied, and divided, just like regular numbers.
Simplifying Surds
To simplify a surd, we need to find the largest perfect square that divides the number inside the square root. We can then take the square root of this perfect square and simplify the expression.
For example, let's simplify the surd √75. We can break down 75 into its prime factors: 75 = 3 × 5 × 5. We can see that 75 has a perfect square factor of 25 (5 × 5). We can then take the square root of 25 and simplify the expression:
√75 = √(3 × 5 × 5) = √(5 × 5) × √3 = 5√3
Simplifying the Given Expression
Now that we have a basic understanding of surds and how to simplify them, let's apply this knowledge to the given expression:
{ \frac{1}{\sqrt{75}+\sqrt{18}-\sqrt{27}} \}
To simplify this expression, we need to simplify each of the surds inside the square roots.
Simplifying √75
We can simplify √75 using the method described above:
√75 = √(3 × 5 × 5) = √(5 × 5) × √3 = 5√3
Simplifying √18
We can simplify √18 using the method described above:
√18 = √(2 × 3 × 3) = √(3 × 3) × √2 = 3√2
Simplifying √27
We can simplify √27 using the method described above:
√27 = √(3 × 3 × 3) = √(3 × 3) × √3 = 3√3
Substituting Simplified Surds
Now that we have simplified each of the surds, we can substitute these simplified expressions back into the original expression:
{ \frac{1}{5\sqrt{3}+3\sqrt{2}-3\sqrt{3}} \}
Combining Like Terms
We can combine like terms in the denominator:
5√3 - 3√3 = 2√3
So, the expression becomes:
{ \frac{1}{2\sqrt{3}+3\sqrt{2}} \}
Rationalizing the Denominator
To rationalize the denominator, we need to multiply the numerator and denominator by the conjugate of the denominator. The conjugate of 2√3 + 3√2 is 2√3 - 3√2.
{ \frac{1}{2\sqrt{3}+3\sqrt{2}} \times \frac{2\sqrt{3}-3\sqrt{2}}{2\sqrt{3}-3\sqrt{2}} \}
Simplifying the Expression
We can simplify the expression by multiplying the numerator and denominator:
{ \frac{2\sqrt{3}-3\sqrt{2}}{(2\sqrt{3})^2-(3\sqrt{2})^2} \}
Using the difference of squares formula, we can simplify the denominator:
{ (2\sqrt{3})^2-(3\sqrt{2})^2 = 4 \times 3 - 9 \times 2 = 12 - 18 = -6 \}
So, the expression becomes:
{ \frac{2\sqrt{3}-3\sqrt{2}}{-6} \}
Final Simplification
We can simplify the expression by dividing the numerator by -6:
{ \frac{2\sqrt{3}-3\sqrt{2}}{-6} = \frac{-1}{3}(\sqrt{3}-\frac{3}{2}\sqrt{2}) \}
Conclusion
In this article, we have simplified the given expression in terms of simplified surds. We have used the method of simplifying surds to break down the expression into its simplest form. We have also used the difference of squares formula to simplify the denominator. The final simplified expression is:
{ \frac{-1}{3}(\sqrt{3}-\frac{3}{2}\sqrt{2}) \}
Introduction
In our previous article, we explored how to simplify surds and express the given expression in terms of simplified surds. In this article, we will answer some of the most frequently asked questions about simplifying surds.
Q: What is a surd?
A: A surd is a mathematical expression that involves the square root of a number. It is often represented as √a, where a is a number.
Q: How do I simplify a surd?
A: To simplify a surd, you need to find the largest perfect square that divides the number inside the square root. You can then take the square root of this perfect square and simplify the expression.
Q: What is the difference of squares formula?
A: The difference of squares formula is a^2 - b^2 = (a + b)(a - b). This formula can be used to simplify expressions that involve the difference of two squares.
Q: How do I rationalize the denominator?
A: To rationalize the denominator, you need to multiply the numerator and denominator by the conjugate of the denominator. The conjugate of a + b is a - b.
Q: What is the conjugate of an expression?
A: The conjugate of an expression is an expression that has the same terms, but with opposite signs. For example, the conjugate of 2√3 + 3√2 is 2√3 - 3√2.
Q: How do I simplify an expression with multiple surds?
A: To simplify an expression with multiple surds, you need to simplify each surd individually and then combine the simplified expressions.
Q: What is the final simplified expression for the given problem?
A: The final simplified expression for the given problem is:
{ \frac{-1}{3}(\sqrt{3}-\frac{3}{2}\sqrt{2}) \}
Q: Can I simplify the expression further?
A: No, the expression cannot be simplified further. It is in its simplest form.
Q: What are some common mistakes to avoid when simplifying surds?
A: Some common mistakes to avoid when simplifying surds include:
- Not finding the largest perfect square that divides the number inside the square root
- Not taking the square root of the perfect square
- Not multiplying the numerator and denominator by the conjugate of the denominator
- Not simplifying each surd individually
Q: How do I practice simplifying surds?
A: You can practice simplifying surds by working through examples and exercises. You can also use online resources and practice tests to help you improve your skills.
Conclusion
In this article, we have answered some of the most frequently asked questions about simplifying surds. We have also provided some tips and resources to help you practice simplifying surds. By following these tips and practicing regularly, you can become proficient in simplifying surds and solving complex mathematical problems.
Additional Resources
- Online resources: Khan Academy, Mathway, and Wolfram Alpha
- Practice tests: Mathway, Khan Academy, and IXL
- Textbooks: "Algebra" by Michael Artin and "Calculus" by Michael Spivak
Final Thoughts
Simplifying surds is an important skill in mathematics, and it requires practice and patience to master. By following the tips and resources provided in this article, you can improve your skills and become proficient in simplifying surds. Remember to always simplify each surd individually and to multiply the numerator and denominator by the conjugate of the denominator. With practice and dedication, you can become a master of simplifying surds.