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Chapter 7: Problem 84

Simplify. $$\sqrt{3 a b}(\sqrt{3 a}+\sqrt{3})$$

Short Answer

Expert verified
3a\sqrt{b} + 3\sqrt{ab}

Step by step solution

01

Distribute the Square Root Expression

Distribute the \(\sqrt{3ab}\) to both terms inside the parentheses. You will have: \[\sqrt{3ab} \cdot \sqrt{3a} + \sqrt{3ab} \cdot \sqrt{3}\]
02

Multiply the Square Roots

Use the property that \(\sqrt{a} \cdot \sqrt{b} = \sqrt{ab}\). This gives: \[\sqrt{3ab \cdot 3a} + \sqrt{3ab \cdot 3}\]
03

Simplify the Expressions Inside the Square Roots

Simplify the products inside each square root: \[\sqrt{9a^2b} + \sqrt{9ab}\]
04

Simplify the Square Roots

Since \(\sqrt{9a^2b} = 3a\sqrt{b}\) and \(\sqrt{9ab} = 3\sqrt{ab}\), you will have: \[3a\sqrt{b} + 3\sqrt{ab}\]
05

Combine Like Terms

There are no like terms to combine, so the simplified expression is: \[3a\sqrt{b} + 3\sqrt{ab}\]

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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 crucial in algebra. It allows us to multiply a single term by each term inside a set of parentheses. In our exercise, we start with \[ \sqrt{3ab}(\sqrt{3a} + \sqrt{3}) \]. We need to distribute the \[ \sqrt{3ab} \] to both terms inside the parentheses. This means: \[ \sqrt{3ab} \cdot \sqrt{3a} + \sqrt{3ab} \cdot \sqrt{3} \]. Using the distributive property makes complex expressions simpler and easier to handle.
Square Roots
Understanding square roots is essential for this exercise. The square root of a number is a value that, when multiplied by itself, gives the original number. For example, \[ \sqrt{9} = 3 \], because \[3 \cdot 3 = 9 \]. In our problem, we deal with expressions like \[\sqrt{3a} \] and \[ \sqrt{3b} \]. Remember that the product of square roots can be combined: \[ \sqrt{a} \cdot \sqrt{b} = \sqrt{ab} \]. This will be helpful in the next steps.
Simplification Steps
To simplify algebraic expressions, follow structured steps:
1. Start by distributing the terms: \[ \sqrt{3ab} \cdot \sqrt{3a} + \sqrt{3ab} \cdot \sqrt{3} \].
2. Multiply the square roots: \[ \sqrt{3ab \cdot 3a} + \sqrt{3ab \cdot 3} \].
3. Simplify the expressions inside the square roots: \[ \sqrt{9a^2b} + \sqrt{9ab} \].
4. Then, simplify the square roots: \[ \sqrt{9a^2b} = 3a\sqrt{b} \], and \[ \sqrt{9ab} = 3\sqrt{ab} \].
By following these clear steps, complex problems become manageable.
Multiplication of Square Roots
When multiplying square roots, use the property \[ \sqrt{a} \cdot \sqrt{b} = \sqrt{ab} \]. This rule helps simplify expressions involving roots. During our exercise, we apply this to: \[ \sqrt{3ab} \cdot \sqrt{3a} = \sqrt{3ab \cdot 3a} = \sqrt{9a^2b} \]. Similarly, \[ \sqrt{3ab} \cdot \sqrt{3} = \sqrt{9ab} \]. Recognizing and applying this multiplication property is key to simplifying expressions involving square roots.

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Most popular questions from this chapter

Solve each problem by writing an equation and solving it. Find the exact answer and simplify it using the rules for radicals. Average annual return. $$r=\left(\frac{S}{P}\right)^{1 / n}-1$$ was used to find the average annual return for an investment. a) Write \(S\) as a function of \(r, P,\) and \(n\) b) Write \(P\) as a function of \(r, S,\) and \(n\)

Solve each equation. $$(t-1)^{-2 / 3}=2$$

Top stock fund. The average annual return \(r\) is a function of the initial investment \(P,\) the number of years \(n,\) and the amount \(S\) that it is worth after \(n\) years: $$r=\left(\frac{S}{P}\right)^{1 / n}-1$$ An investment of \(\$ 10,000\) in the World Precious Minerals Fund in 2001 was worth \(\$ 62,760\) in 2006 (www.money.com). Find the 5 -year average annual return.

Solve each equation. $$x^{-3}=8$$

Solve each equation. $$\sqrt{4 x^{2}}=x+2$$
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