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Chapter 1: Problem 14

Express each in terms of the simplest possible radical. $$\sqrt{\frac{27}{6}}$$

Short Answer

Expert verified
\( \frac{3\sqrt{2}}{2} \)

Step by step solution

01

Simplify Inside the Radical

First, simplify the fraction inside the radical. \( \frac{27}{6} \) can be simplified by dividing both the numerator and the denominator by their greatest common divisor, which is 3. So, \( \frac{27}{6} = \frac{9}{2} \).
02

Apply the Radical to the Simplified Fraction

Next, apply the square root to the simplified fraction: \( \sqrt{\frac{9}{2}} \). This can be rewritten as \( \frac{\sqrt{9}}{\sqrt{2}} \) using the property \( \sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}} \).
03

Simplify the Numerator

Find the square root of the numerator which is 9. Since \( \sqrt{9} = 3 \), the expression becomes \( \frac{3}{\sqrt{2}} \).
04

Rationalize the Denominator

To remove the radical from the denominator, multiply both the numerator and denominator by \( \sqrt{2} \). This results in \( \frac{3\sqrt{2}}{2} \) since \( \sqrt{2} \times \sqrt{2} = 2 \). Thus, the simplest radical form is \( \frac{3\sqrt{2}}{2} \).

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Key Concepts

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

Simplifying Fractions
Simplifying fractions involves reducing them to their simplest form. This makes calculations easier and results more understandable.
To simplify a fraction, identify the greatest common divisor (GCD) of the numerator and the denominator.
  • The GCD is the largest number that can evenly divide both numbers.
  • By dividing the numerator and the denominator by the GCD, you simplify the fraction.
For example, in the fraction \( \frac{27}{6} \), both 27 and 6 can be divided by 3, their GCD.
This step reduces \( \frac{27}{6} \) to \( \frac{9}{2} \). Simplifying fractions is a critical first step in working with more complex expressions.
Rationalizing Denominators
Rationalizing denominators is a process used to eliminate radicals from the denominator of a fraction. Why do we do this?
Fractions are often easier to understand when the denominator is a rational number.
To rationalize a denominator that contains a square root, multiply both the numerator and the denominator by the radical in the denominator.
  • For example, with \( \frac{3}{\sqrt{2}} \), multiply by \( \frac{\sqrt{2}}{\sqrt{2}} \).
  • This makes the denominator a whole number because \( \sqrt{2} \times \sqrt{2} = 2 \).
This process results in \( \frac{3\sqrt{2}}{2} \), a simplified form where the denominator is now rational.
Square Roots
Square roots are a fundamental concept in mathematics, representing a value that, when multiplied by itself, gives the original number.
Understanding how to work with square roots is crucial for simplifying expressions like \( \sqrt{\frac{27}{6}} \).
  • To simplify, notice that \( \sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}} \), allowing separate simplification of the numerator and denominator.
  • For example, the square root of 9 is calculated as \( \sqrt{9} = 3 \).
This transforms \( \sqrt{\frac{9}{2}} \) to \( \frac{3}{\sqrt{2}} \), which is a more manageable form linked directly to square roots and their properties.

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Most popular questions from this chapter

Find all values of \(x\) that make the equation true. $$\left|\frac{6-2 x}{3}\right|+\frac{2}{5}=8$$

Perform the indicated operation and simplify. $$\frac{1}{(x-3)^{2}}-\frac{3}{x-3}$$

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

Determine whether each statement is true for all real numbers \(x\). If the statement is false, then indicate one counterexample, i.e. a value of \(x\) for which the statement is false. $$\frac{1}{x} \leqslant x$$

Solve the inequality. $$9 \leq 8 x-3<11$$
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