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Chapter 5: Problem 48

Perform the indicated operation. Simplify the answer when possible. \(\sqrt{3}+\sqrt{27}\)

Short Answer

Expert verified
The simplified answer is \(4\sqrt{3}\).

Step by step solution

01

Identify the Prime Factors

Begin by identifying the prime factors of the numbers under the square roots. The prime factors of 3 are 1 and 3, while the prime factors of 27 are 3, 3, and 3.
02

Simplify the Radicals

Now, simplify each of the radical terms. The square root of 3 remains as \(\sqrt{3}\), while \(\sqrt{27}\) can be simplified to \(3\sqrt{3}\). This is because \(\sqrt{27}\) can be written as \(\sqrt{3^3}\), which simplifies further to \(3\sqrt{3}\).
03

Combine Like Terms

Final step is to combine like terms. Since both terms are square roots of 3, they can be combined to give \(1\sqrt{3} + 3\sqrt{3} = 4\sqrt{3}\).

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Key Concepts

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

Prime Factorization
Prime factorization is the process of breaking down a number into the prime numbers that multiply together to result in the original number. Prime numbers are numbers greater than 1 that have no divisors other than 1 and themselves. For example, the number 27 can be factored into prime numbers as follows:

27 can be divided by 3, giving 9. Then, 9 can again be divided by 3, leaving 3. Finally, 3 is a prime number. Thus, the prime factors of 27 are 3, 3, and 3, written as \(3^3\).

This process is essential for simplifying square roots because it helps identify which numbers can be pulled out of the square root symbol.
Combining Like Terms
Combining like terms is a fundamental concept in algebra and involves adding or subtracting terms that have the same variables raised to the same power. It makes expressions simpler and easier to work with. In the context of simplifying radicals, like terms are those which have the same radicand (the number inside the square root).

For the expression \(\sqrt{3} + 3\sqrt{3}\), both terms have the radicand 3. Therefore, they can be combined as follows:
  • The coefficient of the first term is 1 (implied), and the second term is 3.
  • Add the coefficients: \(1 + 3 = 4\).
So, \(\sqrt{3} + 3\sqrt{3} = 4\sqrt{3}\). Combining like terms simplifies the expression significantly.
Square Roots
The square root of a number is a value that, when multiplied by itself, gives the original number. Understanding how to simplify square roots can help make calculations easier.

When simplifying square roots using prime factorization, look for pairs of prime factors. For example, in \(\sqrt{27}\), 27 is expressed as \(3^3\) or \(3 \times 3 \times 3\). For every pair of the same number, one can be taken out of the square root. Since there is one pair of 3s in \(3^3\), one 3 comes out:
  • \(\sqrt{27} \rightarrow \sqrt{3^2 \times 3} = 3\sqrt{3}\).
This approach helps streamline expressions and aids in recognizing patterns within algebraic operations.

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Most popular questions from this chapter

Write the first six terms of the geometric sequence with the first term, \(a_{1}\), and common ratio, \(r\). \(a_{1}=-6, r=-5\)

Write a formula for the general term (the nth term) of each geometric sequence. Then use the formula for \(a_{n}\) to find \(a_{7}\), the seventh term of the sequence. \(18,6,2, \frac{2}{3}, \ldots\)

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 \((r \neq 1)\). Determine whether each sequence is arithmetic or geometric. Then use the appropriate formula to find \(S_{10}\), the sum of the first ten terms. \(-15,-9,-3,3, \ldots\)

Find the indicated term for the geometric sequence with first term, \(a_{1}\), and common ratio, \(r\). Find \(a_{40}\), when \(a_{1}=6, r=-1\).

Find the indicated term for the geometric sequence with first term, \(a_{1}\), and common ratio, \(r\). Find \(a_{40}\), when \(a_{1}=1000, r=-\frac{1}{2}\).
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