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Chapter 10: Problem 44

In Exercises \(39-64,\) rationalize each denominator. $$\frac{12}{\sqrt{3 y}}$$

Short Answer

Expert verified
The rationalized form of the given fraction is \(\frac{12\sqrt{3y}}{3y}\)

Step by step solution

01

Identify the square root in the denominator

In this case, the square root in the denominator is \(\sqrt{3y}\)
02

Multiply the numerator and denominator by the conjugate of the denominator

The conjugate of \(\sqrt{3y}\) is \(\sqrt{3y}\) itself. Therefore, multiply both the numerator and denominator of the fraction by \(\sqrt{3y}\) to get \(\frac{12\sqrt{3y}}{3y}\)
03

Simplify the Fraction

Now the denominator is \(3y\), which has no square root. The simplified fraction is \(\frac{12\sqrt{3y}}{3y}\).

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Key Concepts

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

Square Root
A square root is a special value that produces a specific number when multiplied by itself. In simpler terms, if you have a number called "x," its square root is a number "y" such that when you multiply "y" by itself, you get "x." For example, the square root of 9 is 3, because 3 multiplied by itself gives 9. When it comes to simplifying expressions or fractions in mathematics, square roots often appear in denominators, as shown in many exercises.
  • Square roots deal with perfect squares, like 4, 9, 16, etc., where their roots are whole numbers.
  • When not perfect squares, roots can be irrational numbers.
Understanding how to handle square roots is critical for rationalizing denominators. This means you'll often want to remove the square root from the bottom of your fraction to simplify it.
Simplifying Fractions
Simplifying fractions involves rewriting them in their most basic form, so both the numerator and the denominator are reduced to their smallest possible values. You do this by finding the greatest common factor of the numbers in the fraction and dividing both parts by this factor.
  • For example, to simplify the fraction \( \frac{6}{9} \), divide both 6 and 9 by the greatest common factor, 3, resulting in \( \frac{2}{3} \).
  • Simplified fractions make calculations and comparisons easier.
When simplifying fractions with square roots in the denominator, the focus is on making the denominator a rational number. This process is part of the rationalization technique, ensuring no irrational numbers, like square roots, remain in the denominator.
Conjugates
In mathematics, a conjugate is a pair of expressions where two terms (usually involving square roots) are combined differently. For instance, if you have the expression \(a + \sqrt{b}\), its conjugate would be \(a - \sqrt{b}\). The purpose of these is to help eliminate radicals in the denominator.
  • When you multiply a number by its conjugate, you use the formula \(a^2 - b^2\) to achieve a rational number.
  • For example, multiplying \(\frac{1}{\sqrt{3y}}\) by \(\sqrt{3y}/\sqrt{3y}\) results in \(\frac{\sqrt{3y}}{3y}\).
Utilizing conjugates is a powerful tool in algebra for rationalizing denominators. This is vital when dealing with roots, since the goal is to simplify the denominator to make computations easier and expressions clearer.

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Most popular questions from this chapter

Factor: \(y^{2}-6 y+9-25 x^{2}\) (Section 6.5, Example 8)

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. The equations \(\sqrt{x+4}=-5\) and \(x+4=25\) have the same solution set.

Divide using synthetic division: $$\left(4 x^{4}-3 x^{3}+2 x^{2}-x-1\right) \div(x+3)$$ (Section \(5.6,\) Example 5 )

In Exercises \(39-64,\) rationalize each denominator. $$\frac{2 x^{2} y}{\sqrt[5]{4 x^{2} y^{4}}}$$

Describe how to multiply conjugates.
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