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Chapter 8: Problem 59

Rationalize each denominator. Write quotients in lowest terms. $$ \frac{\sqrt{7}+\sqrt{2}}{\sqrt{3}-\sqrt{2}} $$

Short Answer

Expert verified
The rationalized form is \( \root{21} + \root{14} + \root{6} + 2 \).

Step by step solution

01

- Identify the Conjugate

To rationalize the denominator, identify the conjugate of the denominator. The conjugate of \(\root{3} - \root{2}\) is \(\root{3} + \root{2}\).
02

- Multiply Numerator and Denominator by the Conjugate

Multiply both the numerator and the denominator by the conjugate \(\root{3} + \root{2}\): \[ \frac{\root{7} + \root{2}}{\root{3} - \root{2}} \times \frac{\root{3} + \root{2}}{\root{3} + \root{2}}. \] This results in \[ \frac{(\root{7} + \root{2})(\root{3} + \root{2})}{(\root{3} - \root{2})(\root{3} + \root{2})}. \]
03

- Expand the Numerator

Expand the numerator using the distributive property: \[ \root{7}\root{3} + \root{7}\root{2} + \root{2}\root{3} + \root{2}\root{2} = \root{21} + \root{14} + \root{6} + 2. \]
04

- Simplify the Denominator

Simplify the denominator using the difference of squares formula: \[ \root{3}^2 - \root{2}^2 = 3 - 2 = 1. \]
05

- Combine Results

Since the denominator is now 1, the expression simplifies to just the numerator: \[ \root{21} + \root{14} + \root{6} + 2. \]

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Key Concepts

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

Difference of Squares
The difference of squares is a special algebraic pattern used to simplify expressions. It states that for any two numbers, say \(a\) and \(b\), the expression \(a^2 - b^2\) can be factored into \((a + b)(a - b)\).
In the given exercise, the denominator \(\root{3} - \root{2}\) is not yet in a factored form. To rationalize it, we multiply by the conjugate \(\root{3} + \root{2}\).
When we do this multiplication, we utilize the difference of squares formula. Here鈥檚 how it works step-by-step:
  • Original denominator: \(\root{3} - \root{2}\)
  • Conjugate: \(\root{3} + \root{2}\)
  • Multiplying these: \((\root{3} - \root{2})(\root{3} + \root{2})\)
  • Using the formula gives: \((\root{3})^2 - (\root{2})^2\)
  • Which simplifies to: \(3 - 2 = 1\)
As a result, the denominator becomes 1, making the expression easier to handle.
Distributive Property
The distributive property is a fundamental principle in algebra. It allows us to multiply a single term across terms inside parentheses. The property is written as \(a(b + c) = ab + ac\).
In our problem:
The numerator after multiplying by the conjugate is \((\root{7} + \root{2})(\root{3} + \root{2})\). Using the distributive property, we need to distribute each term:
  • \(\root{7} \cdot \root{3}\)
  • \(\root{7} \cdot \root{2}\)
  • \(\root{2} \cdot \root{3}\)
  • \(\root{2} \cdot \root{2}\)
This results in:
  • \(\root{21}\)
  • \(\root{14}\)
  • \(\root{6}\)
  • 2 (since \(\root{2} \cdot \root{2} = 2\))
Combine all these terms to get \(\root{21} + \root{14} + \root{6} + 2\) as the final numerator.
Simplifying Radicals
Simplifying radicals is the process of making a radical expression as simple as possible. This often involves combining like terms or removing a radical from the denominator, as seen in rationalization. Let's look at what was done to simplify the terms:
In our problem, simplifying involves:
  • Combining the radicals in the numerator to simplify the expression.
  • Making the denominator rational, which is 1 in this case.
By multiplying the numerator and denominator with the conjugate, the steps lead to: \(\frac{\root{21} + \root{14} + \root{6} + 2}{1}\).
Finally, since dividing by 1 doesn鈥檛 change the value, the expression simplifies directly to \(\root{21} + \root{14} + \root{6} + 2\).
Always look to combine like terms and ensure all radicals are in their simplest forms.

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Most popular questions from this chapter

If an investment of \(P\) dollars grows to \(A\) dollars in 2 yr, the annual rate of return on the investment is given by $$ r=\frac{\sqrt{A}-\sqrt{P}}{\sqrt{P}} $$ First rationalize the denominator, and then find the annual rate of return (as a percent) if \(\$ 50,000\) increases to \(\$ 54,080.\)

Simplify each radical expression. $$ \sqrt{\frac{8}{3}} \cdot \sqrt{\frac{512}{27}} $$

Simplify each radical. $$ \sqrt[3]{\frac{m^{12}}{8}} $$

Combine like terms. $$ 2 x y+3 x^{2} y-9 x y+8 x^{2} y $$

Simplify each radical. Assume that all variables represent nonnegative real numbers. $$ \sqrt{100 c^{4} d^{6}} $$
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