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Chapter 1: Problem 21

Rationalize the denominator, simplifying if possible. $$\frac{6 \sqrt{7}}{\sqrt{3}}$$

Short Answer

Expert verified
The simplified expression is \(2 \sqrt{21}\).

Step by step solution

01

Identify the Problem

You are given the expression \(\frac{6 \sqrt{7}}{\sqrt{3}}\) and need to rationalize the denominator by removing the square root from the denominator.
02

Multiply by the Conjugate

To rationalize the denominator, multiply both the numerator and the denominator by \(\sqrt{3}\), which is the conjugate of the denominator \(\sqrt{3}\). This gives us:\[\frac{6 \sqrt{7}}{\sqrt{3}} \times \frac{\sqrt{3}}{\sqrt{3}} = \frac{6 \sqrt{21}}{3}.\]
03

Simplify the Result

Divide the numerator and denominator by \(3\) (since \(3\) is a common factor) to simplify the expression:\[\frac{6 \sqrt{21}}{3} = 2 \sqrt{21}.\]

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Key Concepts

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

Understanding the Denominator
In fractions, the denominator is the bottom part that depicts how many parts make a whole. When rationalizing, our goal is to eliminate the square root present in the denominator.
In our exercise example, the denominator was initially \(\sqrt{3}\). Fractions work best with whole numbers in the denominator, as they are easier to work with and understand.
To rationalize, it's necessary to remove any square roots from the denominator, making calculations more straightforward and expressions more pleasing to the eyes.
  • The denominator in a fraction determines the type of portions you are dividing into.
  • Rationalizing helps to transform the denominator into a whole number.
  • This process simplifies further mathematical operations one might need to carry out.
The Role of Square Roots
Square roots play an integral role in mathematics, denoting numbers that produce a specified value when multiplied by themselves. In rationalization, it's crucial to eliminate them from the denominator.
The square root of a number "a" is represented as \(\sqrt{a}\) and essentially asks, "What number, when multiplied by itself, gives a?"
Rationalization involves adjusting the expression such that the denominator no longer retains a square root, simplifying the expression to a more understandable format.
  • Square roots are necessary for expressing irrational numbers precisely.
  • The aim is to shift the square root from the denominator to a simpler position.
  • Expressing square roots correctly can also help in calculations involving powers and roots later on.
The Simplification Process
Simplification is the art of making mathematical expressions easier to handle. The goal is to reduce fractions as much as possible while keeping the meaning intact.
In our worked example, after multiplying by the conjugate \(\sqrt{3}\), we obtained a new expression: \(\frac{6 \sqrt{21}}{3}\). Here, further simplification was needed. By dividing both the numerator and denominator by their common factor, which is 3, we arrived at the ultimate simplified form: \(2 \sqrt{21}\).
  • Always look for common factors that can simplify both the numerator and the denominator.
  • Check whether the numbers are in their simplest form鈥攖his generally leads to easier understanding and operations!
  • Simplification doesn't change the value of the expression; instead, it provides a cleaner and more manageable representation.

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Most popular questions from this chapter

Use scientific notation and the laws of exponents to perform the indicated operations. Give the result in scientific notation rounded to two significant figures. $$\frac{4 \times 10^{4}}{\left(6.4 \times 10^{2}\right)\left(2.5 \times 10^{-5}\right)}$$

Simplify the algebraic fraction. $$\frac{(2 x+h)^{2}-4 x^{2}}{h}$$

Simplify the algebraic fraction. $$\frac{2 a-5}{5-2 a}$$

Rationalize the denominator of each fractional expression. $$\frac{5}{2+x \sqrt{3}}$$

Use the substitution method to solve each pair of simultaneous equations. $$\begin{aligned} &0.4 x+0.3 y=1\\\ &0.25 x+0.1 y=-0.25 \end{aligned}$$
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