Problem 207 - Elementary Algebra

Question

In the following exercises, simplify. $$ 3 \sqrt{128 y^{2}}+4 y \sqrt{162}-8 \sqrt{98 y^{2}} $$

Step-by-Step Solution

Answer

The simplified expression is \(4 y \root{2}\).

Step 1: Simplify Inside the Square Roots

First, simplify the expressions inside the square roots where possible.For \(\root{128 y^{2}}\), factor inside the square root: \[ 128 y^{2} = 2^7 \times y^{2} \]

Step 2: Simplify \(\root{128 y^{2}}\)

Take the square root of the factors in step 1.\[ \root{128 y^{2}} = \root{2^7 y^{2}} = 2^{3.5} y = 8 \root{2} y \]

Step 3: Simplify \(\root{162}\)

Next, simplify \root{162}. Factor inside square root.\[ \root{162} = \root{2 \times 81} = 9 \root{2} \]Thus, \[ 4 y \root{162} = 4 y \times 9 \root{2} = 36 y \root{2} \]

Step 4: Simplify \(\root{98 y^{2}}\)

Factor inside the square root \root{98 y^{2}}.\[ \root{98 y^{2}} = \root{2 \times 49 y^{2}} = 7 y \root{2} \]

Step 5: Combine All Simplified Terms

Now combine all the simplified terms into a single expression.\[ 3 \times 8 y \root{2} + 36 y \root{2} - 8 \times 7 y \root{2} \]

Step 6: Perform Arithmetic Operations

Multiply the constants and add/subtract the simplified terms.\[ 24 y \root{2} + 36 y \root{2} - 56 y \root{2} \]Combine like terms:\[ (24 + 36 - 56) y \root{2} = 4 y \root{2} \]

Key Concepts

Radical Expressions

Simplifying radical expressions involves breaking them down into more manageable parts using factors and roots. It's essential to factorize the number or variable inside the root, which can make simplification easier.
Here’s a general approach to simplifying radical expressions:

• Factorize the number or variable inside the radical.
• Simplify the factors using roots.
• Combine like terms if possible.

This method is used throughout the simplification process.

Square Roots

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Understanding how to manage square roots is fundamental to working with complex algebraic expressions.

Algebraic Simplification

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Mastering algebraic simplification allows you to handle complex equations more easily.

Like Terms

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Recognizing and combining like terms simplifies equations and expressions significantly.