Problem 68 Simplify each expression. $$4 ... [FREE SOLUTION]

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Elementary Algebra

Jerome E. Kaufmann, Karen L. Schwitters

$Math Studyset 91影视 Explanations$ Math

9 Edition

Chapter 9: Problem 68

Simplify each expression. $$4 \sqrt{8}-\frac{6}{\sqrt{2}}$$

Short Answer

Expert verified

The simplified expression is $5\sqrt{2}$.

Step by step solution

Simplify the Square Roots

Simplify the square root of 8. The square root expression $ \sqrt{8} $ can be simplified as follows: \[ \sqrt{8} = \sqrt{4 \times 2} = \sqrt{4} \times \sqrt{2} = 2\sqrt{2} \]Therefore, \[ 4\sqrt{8} = 4 \times 2 \sqrt{2} = 8 \sqrt{2} \]

Rationalize the Denominator

Address the term $-\frac{6}{\sqrt{2}}$ by rationalizing the denominator:Multiply the numerator and the denominator by $ \sqrt{2} $:\[-\frac{6}{\sqrt{2}} \times \frac{\sqrt{2}}{\sqrt{2}} = -\frac{6\sqrt{2}}{2} \ = -3\sqrt{2} \]

Combine Like Terms

Combine the terms derived from Step 1 and Step 2:Now, you have $ 8\sqrt{2} - 3\sqrt{2} $.Combine these like terms:\[ 8\sqrt{2} - 3\sqrt{2} = (8 - 3)\sqrt{2} = 5\sqrt{2} \]

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Key Concepts

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

Square Roots Simplification

When you come across a square root that isn't neat, simplifying it can make your math a lot easier. Square Roots Simplification involves breaking down the number under the square root into smaller factors, particularly perfect squares.
This lets you pull the square root of those smaller factors out of the root. For example, if you have the square root of 8, think of it as the product of two smaller numbers: 4 and 2.
Since the square root of 4 is 2, you can simplify $ \sqrt{8} $ into $ 2\sqrt{2} $. In practical terms, this means expressing numbers under square roots in a way that makes them easier to work with in equations or expressions.

Recognize perfect squares like 4, 9, 16, etc., as they help simplify radicals.
Always aim to condense the radical by extracting these perfect squares out from under the radical sign.

Rationalizing the Denominator

In many cases, we prefer not to have square roots in the denominator of fractions. Rationalizing the Denominator refers to the process of eliminating square roots from the denominator to make expressions easier to handle and compare.
To do this, multiply both the numerator and the denominator by the square root present in the denominator. For example, consider the expression $-\frac{6}{\sqrt{2}}$.
Multiply both parts by $\sqrt{2}$ to eliminate the square root from the denominator: $-\frac{6}{\sqrt{2}} \times \frac{\sqrt{2}}{\sqrt{2}} = -\frac{6\sqrt{2}}{2}$, which simplifies to $-3\sqrt{2}$.

This step makes expressions look neater and is crucial for more complex algebraic manipulations.
Rationalizing helps in comparing and performing arithmetic operations on expressions.

Combining Like Terms

Combining like terms is a fundamental algebra skill that involves simplifying expressions by merging terms that have identical variables and exponents. In the simplified expression, like terms make it much more straightforward to understand and solve.
In our given problem, after rationalizing the denominator and simplifying the square root, you have: $8\sqrt{2} - 3\sqrt{2}$.
These can be combined because they both involve the same radical term $\sqrt{2}$. Rewrite them as a single term like this: $(8 - 3)\sqrt{2}$, which results in $5\sqrt{2}$.

Ensure the terms you are combining have exactly the same radical if dealing with roots.
This process condenses expressions, making them easier to manage and interpret.

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91影视

Simplify each expression. $$4 \sqrt{8}-\frac{6}{\sqrt{2}}$$

Short Answer

Step by step solution

Simplify the Square Roots

Rationalize the Denominator

Combine Like Terms

Key Concepts

Square Roots Simplification

Rationalizing the Denominator

Combining Like Terms

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