Rationalize The Denominator And Simplify: 5 7 + 2 X \frac{5}{7 + 2 \sqrt{x}} 7 + 2 X ​ 5 ​ Assume That The Variable Represents A Real Number.

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Introduction

Rationalizing the denominator is a crucial step in simplifying complex fractions, especially when dealing with expressions involving square roots. In this article, we will focus on rationalizing the denominator and simplifying the given expression: $\frac{5}{7 + 2 \sqrt{x}}$. This process is essential in mathematics, particularly in algebra and calculus, as it helps to eliminate any radical terms in the denominator, making it easier to work with and manipulate the expression.

Understanding the Concept of Rationalizing the Denominator

Rationalizing the denominator involves multiplying both the numerator and the denominator by a specific value, usually a radical expression, to eliminate any radical terms in the denominator. This process is necessary to simplify complex fractions and make them easier to work with. In the given expression, $\frac{5}{7 + 2 \sqrt{x}}$, the denominator contains a square root term, which needs to be rationalized.

Step 1: Identify the Radical Term in the Denominator

The radical term in the denominator is $2 \sqrt{x}$. To rationalize the denominator, we need to multiply both the numerator and the denominator by a value that will eliminate this radical term.

Step 2: Multiply the Numerator and Denominator by the Conjugate of the Denominator

The conjugate of the denominator is $7 - 2 \sqrt{x}$. To rationalize the denominator, we multiply both the numerator and the denominator by this conjugate.

Step 3: Simplify the Expression

After multiplying the numerator and denominator by the conjugate, we get:

$$\frac{5}{7 + 2 \sqrt{x}} \cdot \frac{7 - 2 \sqrt{x}}{7 - 2 \sqrt{x}} = \frac{5(7 - 2 \sqrt{x})}{(7 + 2 \sqrt{x})(7 - 2 \sqrt{x})}$$

Step 4: Simplify the Denominator Using the Difference of Squares Formula

The denominator can be simplified using the difference of squares formula: $(a + b)(a - b) = a^2 - b^2$. In this case, the denominator becomes:

$$(7 + 2 \sqrt{x})(7 - 2 \sqrt{x}) = 7^2 - (2 \sqrt{x})^2 = 49 - 4x$$

Step 5: Simplify the Expression Further

Now that the denominator has been simplified, we can simplify the expression further:

$$\frac{5(7 - 2 \sqrt{x})}{49 - 4x}$$

Conclusion

Rationalizing the denominator and simplifying the given expression $\frac{5}{7 + 2 \sqrt{x}}$ involves multiplying both the numerator and the denominator by the conjugate of the denominator and simplifying the expression further. This process is essential in mathematics, particularly in algebra and calculus, as it helps to eliminate any radical terms in the denominator, making it easier to work with and manipulate the expression.

Final Answer

The final answer is: $\boxed{\frac{5(7 - 2 \sqrt{x})}{49 - 4x}}$

Related Topics

• Rationalizing the denominator
• Simplifying complex fractions
• Algebra
• Calculus
• Square roots
• Radical expressions

Example Use Cases

• Simplifying expressions involving square roots
• Rationalizing denominators in algebra and calculus
• Eliminating radical terms in the denominator
• Simplifying complex fractions

Tips and Tricks

• Always identify the radical term in the denominator
• Multiply the numerator and denominator by the conjugate of the denominator
• Simplify the expression further using the difference of squares formula
• Eliminate any radical terms in the denominator

Common Mistakes

• Failing to identify the radical term in the denominator
• Not multiplying the numerator and denominator by the conjugate of the denominator
• Not simplifying the expression further using the difference of squares formula
• Not eliminating any radical terms in the denominator
  

Introduction

In our previous article, we discussed how to rationalize the denominator and simplify the expression $\frac{5}{7 + 2 \sqrt{x}}$. In this article, we will address some common questions and concerns related to rationalizing the denominator and simplifying complex fractions.

Q&A

Q: What is rationalizing the denominator?

A: Rationalizing the denominator involves multiplying both the numerator and the denominator by a specific value, usually a radical expression, to eliminate any radical terms in the denominator.

Q: Why is rationalizing the denominator important?

A: Rationalizing the denominator is important because it helps to eliminate any radical terms in the denominator, making it easier to work with and manipulate the expression. This is particularly useful in algebra and calculus, where complex fractions are common.

Q: How do I identify the radical term in the denominator?

A: To identify the radical term in the denominator, look for any square root terms. In the expression $\frac{5}{7 + 2 \sqrt{x}}$, the radical term is $2 \sqrt{x}$.

Q: What is the conjugate of the denominator?

A: The conjugate of the denominator is the expression obtained by changing the sign of the radical term. In the expression $\frac{5}{7 + 2 \sqrt{x}}$, the conjugate of the denominator is $7 - 2 \sqrt{x}$.

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, simply multiply the two expressions together. In the expression $\frac{5}{7 + 2 \sqrt{x}}$, we multiply the numerator and denominator by $7 - 2 \sqrt{x}$.

Q: What is the difference of squares formula?

A: The difference of squares formula is $(a + b)(a - b) = a^2 - b^2$. This formula is useful for simplifying expressions involving square roots.

Q: How do I simplify the denominator using the difference of squares formula?

A: To simplify the denominator using the difference of squares formula, simply apply the formula to the expression. In the expression $\frac{5}{7 + 2 \sqrt{x}}$, we simplify the denominator using the difference of squares formula to get $49 - 4x$.

Q: What is the final answer?

A: The final answer is $\boxed{\frac{5(7 - 2 \sqrt{x})}{49 - 4x}}$.

Common Misconceptions

• Many students believe that rationalizing the denominator is only necessary when the denominator contains a square root term. However, rationalizing the denominator is also necessary when the denominator contains a cube root term or any other radical term.
• Some students believe that rationalizing the denominator involves simply multiplying the numerator and denominator by a value that eliminates the radical term. However, this is not always the case, and the value must be the conjugate of the denominator.

Tips and Tricks

• Always identify the radical term in the denominator.
• Multiply the numerator and denominator by the conjugate of the denominator.
• Simplify the expression further using the difference of squares formula.
• Eliminate any radical terms in the denominator.

Example Problems

• Simplify the expression $\frac{3}{2 + \sqrt{5}}$.
• Rationalize the denominator of the expression $\frac{4}{3 - 2 \sqrt{2}}$.
• Simplify the expression $\frac{2}{1 + \sqrt{3}}$.

Solutions

• $\frac{3}{2 + \sqrt{5}} \cdot \frac{2 - \sqrt{5}}{2 - \sqrt{5}} = \frac{3(2 - \sqrt{5})}{2^2 - (\sqrt{5})^2} = \frac{6 - 3 \sqrt{5}}{4 - 5} = \frac{6 - 3 \sqrt{5}}{-1} = -6 + 3 \sqrt{5}$
• $\frac{4}{3 - 2 \sqrt{2}} \cdot \frac{3 + 2 \sqrt{2}}{3 + 2 \sqrt{2}} = \frac{4(3 + 2 \sqrt{2})}{3^2 - (2 \sqrt{2})^2} = \frac{12 + 8 \sqrt{2}}{9 - 8} = \frac{12 + 8 \sqrt{2}}{1} = 12 + 8 \sqrt{2}$
• $\frac{2}{1 + \sqrt{3}} \cdot \frac{1 - \sqrt{3}}{1 - \sqrt{3}} = \frac{2(1 - \sqrt{3})}{1^2 - (\sqrt{3})^2} = \frac{2 - 2 \sqrt{3}}{1 - 3} = \frac{2 - 2 \sqrt{3}}{-2} = -1 + \sqrt{3}$