Simplify: 9 + 8 3 1 − 3 \frac{9+8 \sqrt{3}}{1-\sqrt{3}} 1 − 3 ​ 9 + 8 3 ​ ​

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Introduction

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Simplifying complex expressions is a crucial skill in mathematics, particularly in algebra and trigonometry. In this article, we will focus on simplifying the given expression: $\frac{9+8 \sqrt{3}}{1-\sqrt{3}}$. This expression involves square roots, which can be simplified using various techniques. Our goal is to simplify this expression to its simplest form, making it easier to work with and understand.

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

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To simplify the given expression, we will start by multiplying the numerator and denominator by the conjugate of the denominator. The conjugate of $1-\sqrt{3}$ is $1+\sqrt{3}$. By multiplying both the numerator and denominator by the conjugate, we can eliminate the square root in the denominator.

\frac{9+8 \sqrt{3}}{1-\sqrt{3}} \cdot \frac{1+\sqrt{3}}{1+\sqrt{3}}

Step 2: Expand and Simplify the Expression

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

\frac{(9+8 \sqrt{3})(1+\sqrt{3})}{(1-\sqrt{3})(1+\sqrt{3})}

Step 3: Simplify the Numerator and Denominator

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To simplify the numerator, we will multiply the terms using the distributive property.

(9+8 \sqrt{3})(1+\sqrt{3}) = 9(1+\sqrt{3}) + 8 \sqrt{3}(1+\sqrt{3})

= 9 + 9 \sqrt{3} + 8 \sqrt{3} + 8 \sqrt{3} \sqrt{3}

= 9 + 9 \sqrt{3} + 8 \sqrt{3} + 24

= 33 + 17 \sqrt{3}

To simplify the denominator, we will multiply the terms using the distributive property.

(1-\sqrt{3})(1+\sqrt{3}) = 1^2 - (\sqrt{3})^2

= 1 - 3

= -2

Step 4: Simplify the Expression

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

\frac{33 + 17 \sqrt{3}}{-2}

= -\frac{33}{2} - \frac{17 \sqrt{3}}{2}

Conclusion

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In this article, we simplified the given expression: $\frac{9+8 \sqrt{3}}{1-\sqrt{3}}$. We started by multiplying the numerator and denominator by the conjugate of the denominator, then expanded and simplified the expression. Finally, we simplified the expression by dividing the numerator by the denominator. The simplified expression is: $-\frac{33}{2} - \frac{17 \sqrt{3}}{2}$.

Final Answer

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The final answer is: $\boxed{-\frac{33}{2} - \frac{17 \sqrt{3}}{2}}$

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Introduction

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In our previous article, we simplified the given expression: $\frac{9+8 \sqrt{3}}{1-\sqrt{3}}$. In this article, we will answer some frequently asked questions related to the simplification of this expression.

Q1: What is the conjugate of a binomial expression?

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A1: The conjugate of a binomial expression $a-b$ is $a+b$. For example, the conjugate of $1-\sqrt{3}$ is $1+\sqrt{3}$.

Q2: Why do we multiply the numerator and denominator by the conjugate of the denominator?

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A2: We multiply the numerator and denominator by the conjugate of the denominator to eliminate the square root in the denominator. This is a common technique used in algebra to simplify expressions involving square roots.

Q3: How do we simplify the numerator and denominator after multiplying by the conjugate?

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A3: To simplify the numerator, we multiply the terms using the distributive property. To simplify the denominator, we multiply the terms using the distributive property and then simplify the resulting expression.

Q4: What is the final simplified expression?

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A4: The final simplified expression is: $-\frac{33}{2} - \frac{17 \sqrt{3}}{2}$.

Q5: Why is it important to simplify expressions involving square roots?

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A5: Simplifying expressions involving square roots is important because it makes the expression easier to work with and understand. It also helps to eliminate any errors that may occur when working with complex expressions.

Q6: Can we simplify expressions involving square roots using other techniques?

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A6: Yes, we can simplify expressions involving square roots using other techniques such as rationalizing the denominator or using the Pythagorean identity.

Q7: How do we rationalize the denominator of an expression?

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A7: To rationalize the denominator of an expression, we multiply the numerator and denominator by the conjugate of the denominator. This eliminates the square root in the denominator.

Q8: What is the Pythagorean identity?

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A8: The Pythagorean identity is: $a^2 + b^2 = c^2$, where $a$ and $b$ are the legs of a right triangle and $c$ is the hypotenuse.

Q9: How do we use the Pythagorean identity to simplify expressions involving square roots?

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A9: We use the Pythagorean identity to simplify expressions involving square roots by substituting the values of $a$ and $b$ into the identity and then simplifying the resulting expression.

Q10: Can we use the Pythagorean identity to simplify expressions involving square roots in all cases?

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A10: No, we cannot use the Pythagorean identity to simplify expressions involving square roots in all cases. The Pythagorean identity is only applicable to expressions involving square roots of perfect squares.

Conclusion

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In this article, we answered some frequently asked questions related to the simplification of the expression: $\frac{9+8 \sqrt{3}}{1-\sqrt{3}}$. We covered topics such as the conjugate of a binomial expression, multiplying the numerator and denominator by the conjugate, and using the Pythagorean identity to simplify expressions involving square roots.

Final Answer

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The final answer is: $\boxed{-\frac{33}{2} - \frac{17 \sqrt{3}}{2}}$