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Chapter 5: Problem 25

Use the distributive property to simplify the radical expressions. \(\sqrt{3}(5+\sqrt{3})\)

Short Answer

Expert verified
The simplified form of the radical expression \(\sqrt{3}(5+\sqrt{3})\) is \(5\sqrt{3}+3\).

Step by step solution

01

Distribute the Radicand

Distribute \(\sqrt{3}\) to each component inside the parentheses. This gives \(5\sqrt{3} + 3\). The distributive property states that for all real numbers 'a', 'b', and 'c', \(a(b+c) = ab+ ac\). Here, 'a' is \(\sqrt{3}\), 'b' is '5', and 'c' is \(\sqrt{3}\). Thus apply the distributive property to \(\sqrt{3}(5+\sqrt{3})\) as follows: \[\sqrt{3}*5 + \sqrt{3}*\sqrt{3}\]
02

Simplify the Radical Expression

The first term becomes \(5\sqrt{3}\) and the second term becomes 3 (because \(\sqrt{3}*\sqrt{3} = 3\)). This leads to the simplified form \(5\sqrt{3}+3\)

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Key Concepts

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

Radical Expressions
Radical expressions involve roots, most commonly square roots. The square root of a number is a value which, when multiplied by itself, gives the original number. In expressions, we represent the square root using the radical symbol \( \sqrt{} \). For instance, \( \sqrt{9} = 3 \) because \( 3 \times 3 = 9 \). However, when you have more complex expressions such as \( \sqrt{3}(5+\sqrt{3}) \), it gets a bit more intricate. The radicals interact with all terms within parentheses, and you often use other algebraic rules, like the distributive property, to simplify them.
Simplification
Simplification involves reducing an expression to its simplest form. This usually means removing any unnecessary complexity while ensuring the expression retains its original value. In the context of radical expressions, simplification often requires you to distribute radicals over terms inside parentheses. You apply operations like multiplication and look for opportunities to reduce terms, such as multiplying radicals together. In this case, the expression \( \sqrt{3}(5+\sqrt{3}) \) simplifies to \( 5\sqrt{3} + 3 \), using the distributive property by expanding \( \sqrt{3} \times 5 + \sqrt{3} \times \sqrt{3} \).
Algebra
Algebra is a branch of mathematics dealing with symbols and the rules for manipulating these symbols. It is foundational for solving equations and describes relationships between numbers. One common tool in algebra is the distributive property, which helps simplify expressions by distributing one term across the terms in parentheses. This rule states \( a(b+c) = ab + ac \), an essential process in transforming complex expressions into a simpler form. For instance, it allowed us to simplify \( \sqrt{3}(5+\sqrt{3}) \) by expanding it into \( 5\sqrt{3} + 3 \).
Real Numbers
Real numbers include all the numbers you can find on a number line. This covers both rational numbers, like fractions and integers, and irrational numbers, such as \( \sqrt{2} \) and \( \pi \). Real numbers are significant in the context of radical expressions because they provide a broad set of values that fit within these expressions. Whether dealing with whole numbers, fractions, or roots, algebra manipulations like simplification via the distributive property depend on all numbers being real to ensure valid operations. This ensures all terms in an expression such as \( \sqrt{3}(5+\sqrt{3}) \) remain legitimate under the rules of arithmetic.

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Most popular questions from this chapter

The sum, \(S_{n}\), of the first \(n\) terms of an arithmetic sequence is given by$$S_{n}=\frac{n}{2}\left(a_{1}+a_{n}\right),$$in which \(a_{1}\) is the first term and \(a_{n}\) is the nth term. The sum, \(S_{n}\), of the first \(n\) terms of a geometric sequence is given by$$S_{n}=\frac{a_{1}\left(1-r^{n}\right)}{1-r},$$ in which \(a_{1}\) is the first term and \(r\) is the common ratio \((r \neq 1)\). In Exercises 115-122, determine whether each sequence is arithmetic or geometric. Then use the appropriate formula to find \(S_{10}\), the sum of the first ten terms. \(4,10,16,22, \ldots\)

Find the indicated term for the geometric sequence with first term, \(a_{1}\), and common ratio, \(r\). Find \(a_{100}\), when \(a_{1}=50, r=1\).

Find the indicated term for the geometric sequence with first term, \(a_{1}\), and common ratio, \(r\). Find \(a_{5}\), when \(a_{1}=4, r=3\).

Find the indicated term for the geometric sequence with first term, \(a_{1}\), and common ratio, \(r\). Find \(a_{20}\), when \(a_{1}=2, r=2\).

In Exercises 133-134, you will develop geometric sequences that model the population growth for California and Texas, the two most populated U.S. states. The table shows the population of California for 2000 and 2010 , with estimates given by the U.S. Census Bureau for 2001 through \(2009 .\) $$ \begin{aligned} &\begin{array}{|l|c|c|c|c|c|} \hline \text { Year } & \mathbf{2 0 0 0} & \mathbf{2 0 0 1} & \mathbf{2 0 0 2} & \mathbf{2 0 0 3} & \mathbf{2 0 0 4} \\ \hline \begin{array}{l} \text { Population } \\ \text { in millions } \end{array} & 33.87 & 34.21 & 34.55 & 34.90 & 35.25 \\ \hline \end{array}\\\ &\begin{array}{|l|c|c|c|c|c|c|} \hline \text { Year } & \mathbf{2 0 0 5} & \mathbf{2 0 0 6} & \mathbf{2 0 0 7} & \mathbf{2 0 0 8} & \mathbf{2 0 0 9} & \mathbf{2 0 1 0} \\ \hline \begin{array}{l} \text { Population } \\ \text { in millions } \end{array} & 35.60 & 36.00 & 36.36 & 36.72 & 37.09 & 37.25 \\ \hline \end{array} \end{aligned} $$ a. Divide the population for each year by the population in the preceding year. Round to two decimal places and show that California has a population increase that is approximately geometric. b. Write the general term of the geometric sequence modeling California's population, in millions, \(n\) years after \(1999 .\) c. Use your model from part (b) to project California's population, in millions, for the year 2020 . Round to two decimal places.
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