Multiply: [tex](\sqrt{10}+2 \sqrt{8})(\sqrt{10}-2 \sqrt{8})[/tex]
Multiply: [tex](\sqrt{2 x^3}+\sqrt{12 x})(2 \sqrt{10 x^5}+\sqrt{6 x^2})[/tex]
Answer (2)
The solution to the first expression is -22, while the simplified form of the second expression is 4 x 4 5 + 2 x 2 3 x + 4 x 3 30 + 6 x 2 x .
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The first expression ( 10 + 2 8 ) ( 10 − 2 8 ) is simplified using the difference of squares formula: a 2 − b 2 , resulting in − 22 .
The second expression ( 2 x 3 + 12 x ) ( 2 10 x 5 + 6 x 2 ) is expanded using the distributive property.
Each term in the expanded expression is simplified by extracting perfect squares from the radicals.
The final simplified expression is 4 x 4 5 + 2 x 2 3 x + 4 x 3 30 + 6 x 2 x .
− 22 and 4 x 4 5 + 2 x 2 3 x + 4 x 3 30 + 6 x 2 x
Explanation
Problem Overview We are given two multiplication problems involving binomial radical expressions. We need to find the correct product for each one.
First Problem Setup The first problem is ( 10 + 2 8 ) ( 10 − 2 8 ) . This is in the form of ( a + b ) ( a − b ) , which simplifies to a 2 − b 2 . Here, a = 10 and b = 2 8 .
First Problem Solution Applying the difference of squares formula, we have: ( 10 ) 2 − ( 2 8 ) 2 = 10 − 4 ( 8 ) = 10 − 32 = − 22
Second Problem Setup The second problem is ( 2 x 3 + 12 x ) ( 2 10 x 5 + 6 x 2 ) . We will use the distributive property (FOIL method) to expand this product.
Expanding the Second Problem Expanding the product, we get: 2 x 3 ∗ 2 10 x 5 + 2 x 3 ∗ 6 x 2 + 12 x ∗ 2 10 x 5 + 12 x ∗ 6 x 2 Simplifying each term: 2 20 x 8 + 12 x 5 + 2 120 x 6 + 72 x 3
Simplifying the Second Problem Further simplifying by extracting perfect squares from the radicals: 2 4 ∗ 5 x 8 + 4 ∗ 3 x 4 ∗ x + 2 4 ∗ 30 x 6 + 36 ∗ 2 x 2 ∗ x = 4 x 4 5 + 2 x 2 3 x + 4 x 3 30 + 6 x 2 x
Final Answer Therefore, the solution to the first problem is -22, and the solution to the second problem is 4 x 4 5 + 2 x 2 3 x + 4 x 3 30 + 6 x 2 x .
Examples
Multiplying radical expressions is a fundamental skill in algebra and is used in various fields such as physics and engineering. For instance, when calculating the impedance of an electrical circuit or determining the area of a complex shape, you might encounter expressions involving radicals that need to be simplified through multiplication. Understanding how to manipulate these expressions allows for accurate calculations and problem-solving in real-world applications.
