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Chapter 0: Problem 37

Rationalize the numerator or denominator and simplify. $$ \frac{2 x}{5-\sqrt{3}} $$

Short Answer

Expert verified
The simplified and rationalized form of the given expression is \(\frac{5x + x\sqrt{3}}{11}\)

Step by step solution

01

Identify the Conjugate

Identify the conjugate of the denominator. Here, the denominator is \(5-\sqrt{3}\), hence its conjugate is \(5 + \sqrt{3}\). This is achieved by changing the sign in between the terms.
02

Multiply by Conjugate

Multiply the fraction by the conjugate of the denominator divided by itself, which means multiply both the numerator and denominator by \(5 + \sqrt{3}\). The expression becomes: \(\frac{2x(5 + \sqrt{3})}{(5 - \sqrt{3})(5 + \sqrt{3})}\)
03

Simplify the Expression

Simplify the expression by calculating numerator and denominator separately. Distribute \(2x\) in the numerator: \(10x + 2x\sqrt{3}\). Use the difference of squares formula in the denominator: \(a^2 - b^2 = (a - b)(a + b)\), where \(a=5\) and \(b=\sqrt{3}\). This equals \(5^2 - (\sqrt{3})^2 = 25 - 3 = 22\). Thus the expression simplifies to \(\frac{10x + 2x\sqrt{3}}{22}\)
04

Further Simplify the Expression

We can simplify this expression further by dividing every term by the common factor 2 getting: \(\frac{5x + x\sqrt{3}}{11}\)

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Conjugates
When dealing with expressions that have terms like \( a + b\sqrt{n} \) or \( a - b\sqrt{n} \) in the numerator or denominator, their conjugates are simple to identify. The conjugate is found by changing the sign between these two terms. For example, if you start with \( 5 - \sqrt{3} \), its conjugate will be \( 5 + \sqrt{3} \). Conjugates are crucial in rationalizing denominators as they help eliminate the radical by relying on the binomial multiplication trick.
  • Conjugate pairs multiply to become a real number, often simplifying the radical part.
  • Using conjugates transforms irrational denominators into something manageable.
These properties make conjugates especially useful in simplifying expressions to a more "rational" form.
Difference of Squares
The difference of squares is a mathematical identity that states: \[ a^2 - b^2 = (a-b)(a+b) \]. It is often used in the context of conjugates, particularly in rationalizing denominators. When you multiply a number by its conjugate, you effectively implement this identity. For example, multiplying \( 5 - \sqrt{3} \) by \( 5 + \sqrt{3} \):
  • Apply the difference of squares: \( (5)^2 - (\sqrt{3})^2 \).
  • This results in \( 25 - 3 = 22 \).
This trick efficiently eliminates the square root from the denominator, leaving us with a simpler expression that doesn't have any radicals. This understanding of the difference of squares is pivotal for simplifying expressions and making them easier to work with.
Simplification of Algebraic Expressions
Once you've rationalized the denominator using conjugates and the difference of squares, your expression often simplifies further. Simplification involves reducing fractions by handling both numerator and denominator appropriately. Here's how:
  • Distribute terms in the numerator first. For example, \( 2x(5 + \sqrt{3}) \) expands to \( 10x + 2x\sqrt{3} \).
  • Once the radicals are gone from the denominator, check for any common factors that can be cancelled from the numerator and denominator. This can simplify the expression even further.
  • The final expression should be in the simplest form possible; in our example, we reduced \( \frac{10x + 2x\sqrt{3}}{22} \) to \( \frac{5x + x\sqrt{3}}{11} \), by dividing everything by 2.
This process makes algebraic expressions cleaner and more concise, emphasizing clarity and ease of calculation.

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Most popular questions from this chapter

Use the Rational Zero Theorem as an aid in finding all real zeros of the polynomial. $$ x^{3}-3 x^{2}-3 x-4 $$

Perform the indicated operations and rationalize as needed. $$ \frac{\frac{\sqrt{x^{2}+1}}{x^{2}}-\frac{1}{x \sqrt{x^{2}+1}}}{x^{2}+1} $$

find the domain of the given expression. $$ \sqrt{x^{2}+3} $$

Find the interval (or intervals) on which the given expression is defined. $$ \sqrt{3 x^{2}-10 x+3} $$

simplify the expression. $$ \left(\frac{12 s^{2}}{9 s}\right)^{3} $$
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