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Chapter 5: Problem 47

Perform the indicated operation. Simplify the answer when possible. \(\sqrt{5}+\sqrt{20}\)

Short Answer

Expert verified
The simplified expression is \( 3\sqrt{5} \).

Step by step solution

01

Identify the square roots

The given expression includes two square roots, we need to add: \( \sqrt{5} \) and \( \sqrt{20} \)
02

Simplify the square roots

The square root of 20 can be simplified. The number 20 is made up by multiplying 4 (a perfect square) by 5. So, \( \sqrt{20} = \sqrt{4 \cdot 5} = \sqrt{4} \cdot \sqrt{5} = 2\sqrt{5} \). The original expression now simplifies to \( \sqrt{5} + 2\sqrt{5} \).
03

Combine like terms

Now, we just have to combine like terms. Both terms under the square root are the same, so we can simply add the numbers in front of them: \( \sqrt{5} + 2\sqrt{5} = 3\sqrt{5} \).

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Key Concepts

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

Radicals Operation
Understanding how to operate with radicals鈥攍ike square roots鈥攊s critical for solving a variety of algebra problems. A radical can be simplified by factoring out perfect squares, which makes it easier to add, subtract, multiply, or divide. The basic rule for the addition and subtraction of radicals is that you can only combine like terms, which are radicals with the same value under the square root.
Combine Like Terms
Combining like terms involves finding terms in an algebraic expression that have the exact same variables raised to the exact same power, then combining them into a single term through addition or subtraction. When it comes to radicals, like terms will have the same radical part, for instance, two terms that are both multiples of \( \sqrt{5} \). By combining these terms, you simplify the expression and find a solution that is easier to understand and to work with in subsequent calculations.
Perfect Squares
Perfect squares are numbers that have integer square roots. Examples include 1, 4, 9, 16, 25, etc. Knowing your perfect squares is useful when simplifying square roots. If the number under the square root is a perfect square, the root simplifies completely. For example, since 4 is a perfect square, \( \sqrt{4} \) simplifies to 2. When you encounter a composite number under a square root, like 20, you'd look to factor it into a perfect square and another number鈥攍ike 4 \(\times\) 5鈥攕o you can simplify the square root of the perfect square and make the entire expression cleaner and simpler to work with.

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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 arithmetic sequence. Then use the formula for \(a_{n}\) to find \(a_{20}\), the 20 the term of the sequence. \(a_{1}=-20, d=-4\)

A professional baseball player signs a contract with a beginning salary of \(\$ 3,000,000\) for the first year with an annual increase of \(4 \%\) per year beginning in the second year. That is, beginning in year 2 , the athlete's salary will be \(1.04\) times what it was in the previous year. What is the athlete's salary for year 7 of the contract? Round to the nearest dollar.

Determine whether each sequence is arithmetic or geometric. Then find the next two terms. \(\frac{1}{2}, 1, \frac{3}{2}, 2, \ldots\)

Find the indicated term for the geometric sequence with first term, \(a_{1}\), and common ratio, \(r\). Find \(a_{8}\), when \(a_{1}=12, r=\frac{1}{2}\).
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