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Chapter 9: Problem 67

Simplify each expression. $$-3 \sqrt{6}-\frac{5 \sqrt{2}}{\sqrt{3}}$$

Short Answer

Expert verified
The simplified expression is \(-\frac{14 \sqrt{6}}{3}\).

Step by step solution

01

Simplify the Second Term

To simplify the term \(-\frac{5 \sqrt{2}}{\sqrt{3}}\), multiply the numerator and the denominator by \(\sqrt{3}\). This helps us remove the square root from the denominator:\[-\frac{5 \sqrt{2}}{\sqrt{3}} \times \frac{\sqrt{3}}{\sqrt{3}} = -\frac{5 \sqrt{2 \cdot 3}}{\sqrt{3 \cdot 3}} = -\frac{5 \sqrt{6}}{3}\]
02

Combine Like Terms

Now we have the expression:\(-3 \sqrt{6} - \frac{5 \sqrt{6}}{3}\).Since both terms contain \(\sqrt{6}\), they can be combined. First, convert \( -3 \sqrt{6}\) to a fraction with a denominator of 3:\[-3 \sqrt{6} = -\frac{9 \sqrt{6}}{3}\]
03

Simplify the Combined Expression

Combine the two fractions:\[-\frac{9 \sqrt{6}}{3} - \frac{5 \sqrt{6}}{3} = \frac{-9 \sqrt{6} - 5 \sqrt{6}}{3} = \frac{-14 \sqrt{6}}{3}\]
04

Write the Final Answer

The simplified expression is \[-\frac{14 \sqrt{6}}{3}\].

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Key Concepts

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

Square Roots
Square roots are fundamental in mathematics, especially when simplifying expressions. A square root of a number is a value that, when multiplied by itself, gives the original number. Square roots are often represented by the radical symbol \( \sqrt{} \). For example, the square root of 16 is 4, since \( 4 \times 4 = 16 \).
  • The expression \( \sqrt{6} \) signifies a number which, when squared, results in 6.
  • Similarly, \( \sqrt{2} \) and \( \sqrt{3} \) represent the square roots of 2 and 3, respectively.
Understanding square roots is key when working with irrational numbers, which cannot be expressed as a simple fraction. In expressions where you have terms like \( \frac{5 \sqrt{2}}{\sqrt{3}} \), the goal is often to simplify these to a more standard form, which leads us to rationalizing denominators.
Combining Like Terms
Combining like terms is a crucial step in simplifying algebraic expressions. This involves grouping together terms that have the same variables raised to the same power. For instance, in expressions like \( -3 \sqrt{6} - \frac{5 \sqrt{6}}{3} \), both terms contain \( \sqrt{6} \), enabling them to be combined.
  • The first term is \( -3 \sqrt{6} \), which can be rewritten with a common denominator for simplicity, such as \( -\frac{9 \sqrt{6}}{3} \).
  • Adding or subtracting like terms essentially combines their coefficients. In this example, the coefficients \(-9\) and \(-5\) of \( \sqrt{6} \) are added to get \(-14\).
This results in a simplified expression \( \frac{-14 \sqrt{6}}{3} \), which highlights the importance of identifying and combining like terms to reduce complexity.
Rationalizing Denominators
Rationalizing the denominator is a technique used to eliminate square roots in the denominator of a fraction. This procedure makes the expression easier to work with and is often required to reach the simplest form. The method involves multiplying the numerator and the denominator by a suitable radical.
  • In the original expression \( -\frac{5 \sqrt{2}}{\sqrt{3}} \), rationalizing involves multiplying both the numerator and denominator by \( \sqrt{3} \), giving us \( -\frac{5 \sqrt{6}}{3} \).
  • The rule here is to multiply by a form of one, like \( \frac{\sqrt{3}}{\sqrt{3}} \), which keeps the value of the expression unchanged.
By applying this technique, the radical is removed from the denominator, allowing for further simplification of the algebraic expression. This results in calculations that are more straightforward and aligns with conventional mathematical practices.

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