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Chapter 10: Problem 4

Rationalize the denominator of each expression. $$\frac{1}{\sqrt{6}}$$

Short Answer

Expert verified
The short answer to the question is: \(\frac{\sqrt{6}}{6}\).

Step by step solution

01

Identify the conjugate of the denominator

First, we need to find the conjugate of the denominator. The conjugate of \(\sqrt{6}\) is just itself, as there is no subtraction or addition involved.
02

Multiply by the conjugate

Now, we will multiply both the numerator and the denominator by the conjugate of \(\sqrt{6}\), which is also \(\sqrt{6}\). \[\frac{1}{\sqrt{6}} \times \frac{\sqrt{6}}{\sqrt{6}}\]
03

Simplify the expression

We will now simplify the expression by multiplying the terms. \[\frac{\sqrt{6}}{(\sqrt{6})(\sqrt{6})}\] \(\sqrt{6}\) multiplied by \(\sqrt{6}\) is just 6, so we have: \[\frac{\sqrt{6}}{6}\]
04

Write the final answer

Our expression with the rationalized denominator is \(\frac{\sqrt{6}}{6}\).

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Key Concepts

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

Conjugate in Mathematics
In mathematics, the concept of a conjugate is quite essential, especially when dealing with numbers that involve radicals or complex numbers. To put it simply, a conjugate is used to eliminate radicals from the denominator of a fraction. When dealing with expressions that involve square roots, the conjugate is especially useful in rationalization.
  • The **conjugate** of an expression in the form of \(a + \sqrt{b}\) is \(a - \sqrt{b}\).
  • When the expression is a single square root like \(\sqrt{a}\), its conjugate is itself.
Using the conjugate helps to multiply the terms in such a way that the denominator becomes a rational number. This is done by eliminating the square root through multiplication, benefiting further calculations. In our exercise, we multiply both the numerator and denominator by \(\sqrt{6}\) to rationalize it.
Simplifying Radicals
Simplifying radicals involves making a radical expression as simple as possible. This might mean breaking down a square root into its simplest form, or making the expression easier to work with by getting rid of the radical in the denominator.
  • When simplifying \(\sqrt{a} \times \sqrt{a}\), this results in \(a\), because the square root and squaring are inverse operations.
  • Simplifying radicals often involves factoring under the square root to simplify the number to its lowest terms.
In the original exercise, the multiplication of \(\sqrt{6}\) by \(\sqrt{6}\) results in 6, which shows how simplification can transform a radical expression into a simpler fraction, facilitating easier manipulation of the expression in further algebraic processes.
Algebraic Fractions
Algebraic fractions are fractions where the numerator and/or the denominator are algebraic expressions. These can include numbers, variables, and radicals. Working with algebraic fractions often requires simplifying, factoring, or rationalizing to make them easier to work with.
  • **Rationalizing** algebraic fractions often involves multiplying both the top and bottom by a term that will eliminate radicals from the denominator.
  • **Simplified form** of an algebraic fraction is reached when the numerator and denominator have no common factors, except 1.
In this exercise, rationalizing the denominator of \(\frac{1}{\sqrt{6}}\) transforms it into the algebraic fraction \(\frac{\sqrt{6}}{6}\), maintaining equivalent value while facilitating operations without radicals in the denominator. It's super useful in simplifying complex expressions in broader algebraic contexts.

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Most popular questions from this chapter

For shallow water waves, the wave velocity is given by $$c=\sqrt{g H}$$ (IMAGE CAN NOT COPY) where \(g\) is the acceleration due to gravity \(\left(32 \mathrm{ft} / \mathrm{sec}^{2}\right)\) and \(H\) is the depth of the water (in feet). a) Find the velocity of a wave in 8 ft of water. b) Solve the equation for \(H\)

Rationalize the denominator of each expression. Assume all variables represent positive real numbers. $$\frac{9}{\sqrt[3]{25}}$$

Simplify completely. $$\frac{30-18 \sqrt{5}}{4}$$

Rationalize the denominator of each expression. Assume all variables represent positive real numbers. $$\sqrt[5]{\frac{7}{4}}$$

Simplify completely. Assume all variables represent positive real numbers. $$\sqrt[4]{x^{20}}$$
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