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Chapter 7: Problem 48

Simplify the products. Give exact answers. $$2 \sqrt{5}(\sqrt{3}+3 \sqrt{5})$$

Short Answer

Expert verified
The simplified expression is \(2 \sqrt{15} + 30\).

Step by step solution

01

Distribute the Expression

Distribute the term outside the parentheses, which is the constant and the radical. In other words, multiply each term inside the parentheses by the term outside. So, we distribute: \[2 \times \sqrt{5} \times (\sqrt{3} + 3 \sqrt{5}) = 2 \sqrt{5} \times \sqrt{3} + 2 \sqrt{5} \times 3 \sqrt{5}\]
02

Simplify Each Term

Simplify each product separately. Remember that the product of square roots can be combined under a single radical as the product of the radicands.First term: \[2 \sqrt{5} \times \sqrt{3} = 2 \sqrt{15}\]Second term: \[2 \sqrt{5} \times 3 \sqrt{5} = 2 \times 3 \sqrt{5 \times 5} = 6 \sqrt{25}= 6 \times 5 = 30\]
03

Combine the Simplified Terms

Add the simplified terms from Step 2 together: \[2 \sqrt{15} + 30\]

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Key Concepts

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

distributive property
The distributive property is a key math concept. It lets you multiply a term outside a parenthesis by each term inside the parenthesis. This means when you're working with expressions like \(2 \sqrt{5}(\sqrt{3}+3 \sqrt{5})\), you follow this property. You distribute \(2 \sqrt{5}\) to both \(\sqrt{3}\) and \(3 \sqrt{5}\). So, by applying the distributive property, you get two separate multiplication problems: \[2 \sqrt{5} \times \sqrt{3} + 2 \sqrt{5} \times 3 \sqrt{5}\]. This might seem a bit tricky but take one step at a time. Distribute the term outside to each term inside the parentheses to ensure every part is included.
multiplication of radicals
Working with radicals can be easy if you understand the rules. When multiplying radicals, you can combine them under a single radical. For example, in the expression \(2 \sqrt{5} \times \sqrt{3}\), you multiply the numbers inside the radicals (called radicands). So, \(\sqrt{5} \times \sqrt{3} = \sqrt{15}\). This gives us the term \(2 \sqrt{15}\). For the second part, \[2 \sqrt{5} \times 3 \sqrt{5}\]: \sqrt{5} \times \sqrt{5} = \sqrt{25} = 5\, so this becomes 2 \times 3 \times 5, which simplifies to 30.
combining like terms
Combining like terms is crucial in simplifying expressions. Like terms are terms whose variable parts are the same. After applying distributive property and simplifying multiplication of radicals, we're left with: \(2 \sqrt{15} + 30\). Here, \(2 \sqrt{15}\) cannot be combined with 30 because they are not like terms. One term has a radical, while the other is a whole number. So, the final simplified form of the expression is: \[2 \sqrt{15} + 30\]. Always double-check to ensure all terms are combined correctly. Understanding how to combine like terms helps in keeping your work clear and accurate.

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Most popular questions from this chapter

Simplify. $$\frac{4}{2 \sqrt{3}}+\frac{1}{\sqrt{5}}$$

Use a calculator to find approximate solutions to the following equations. Round your answers to three decimal places. $$x^{2}=3.24$$

Solve each problem by writing an equation and solving it. Find the exact answer and simplify it using the rules for radicals. Find the length of the side of a square sign whose area is 50 square feet.

Solve each equation. $$(5 x)^{9}=0$$

Discussion Is \(\sqrt{a}+\sqrt{b}=\sqrt{a+b}\) for all values of \(a\) and \(b\) ?
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