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Chapter 0: Problem 41

In Exercises \(33-44,\) add or subtract terms whenever possible. $$3 \sqrt{18}+5 \sqrt{50}$$

Short Answer

Expert verified
The simplified form of \(3 \sqrt{18} + 5 \sqrt{50}\) is \(34\sqrt{2}\).

Step by step solution

01

Simplify the square root terms

First, simplify each of the square root terms separately. To simplify a square root, look for perfect square factors of the number under the square root sign. For example, for \(\sqrt{18}\), the perfect square factor is \(9\), as \(18=9*2\). Therefore, \(\sqrt{18}\) simplifies to \(3*\sqrt{2}\). Similarly, \(\sqrt{50}\) simplifies to \(5*\sqrt{2}\) since \(50 = 25 * 2\).
02

Perform multiplication outside the square roots

Next, perform the multiplication outside each of the square roots. The first term becomes \(3*3*\sqrt{2} = 9*\sqrt{2}\). The second term becomes \(5*5*\sqrt{2} = 25*\sqrt{2}\). So our expression simplifies to \(9 \sqrt{2} + 25 \sqrt{2}\).
03

Combine like terms

Finally, combine like terms by adding together. Both terms have the same \(\sqrt{2}\) factor, so they are like terms. Add the coefficients to get \(34 \sqrt{2}\).

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Key Concepts

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

Adding Square Roots
When dealing with square roots, it's important to remember you can only directly add them if they have the same radicand (the number inside the square root). If not, they need to be simplified first. In the expression given, both square roots were simplified to have the same radicand before adding them. Initially, we had different square roots: \(3 \sqrt{18}\) and \(5 \sqrt{50}\).

Simplification led them to have the same radicand \(\sqrt{2}\), allowing them to be combined. The key point when adding square roots is ensuring they have the same radicand to add the coefficients directly and simplify the expression easily.
Perfect Square Factors
Finding perfect square factors is crucial when simplifying square roots. A perfect square is a number that is the square of an integer, such as 1, 4, 9, 16, and so on. To simplify a square root, we have to identify the largest perfect square that can evenly divide the number inside the square root.

  • For \(\sqrt{18}\), the factors are 1, 2, 3, 6, 9, and 18. The largest perfect square here is 9.
  • For \(\sqrt{50}\), the factors are 1, 2, 5, 10, 25, and 50. The largest perfect square is 25.

By taking these perfect square factors, we can simplify the expression into forms that are easier to manipulate, such as \(3\sqrt{2}\) and \(5\sqrt{2}\), by extracting the perfect square. This reduces complexity and unifies the terms.
Combining Like Terms
Once you simplify square roots and find they share a common radicand, you can combine them just like you would combine like terms in algebra. In this problem, after simplification, both terms had \(\sqrt{2}\) as the radicand: \(9\sqrt{2}\) and \(25\sqrt{2}\).

The principle of combining like terms involves adding or subtracting the coefficients while keeping the square root part unchanged. Thus, the coefficients 9 and 25 add up to make 34, so the expression \(9\sqrt{2} + 25\sqrt{2}\) is simplified to \(34\sqrt{2}\).

This shows how working with like terms in expressions, even when involving square roots, follows the same straightforward process as basic algebraic terms.

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Most popular questions from this chapter

Perform the indicated operations. Simplify the result, if possible. $$\left(4-\frac{3}{x+2}\right)\left(1+\frac{5}{x-1}\right)$$

Explain how to add rational expressions having no common factors in their denominators. Use \(\frac{3}{x+5}+\frac{7}{x+2}\) in your explanation.

Will help you prepare for the material covered in the next section. Jane's salary exceeds Jim's by \(\$ 150\) per week. If \(x\) represents Jim's weekly salary, write an algebraic expression that models Jane's weekly salary.

Describe ways in which solving a linear inequality is similar to solving a linear equation.

In Exercises \(133-136\), determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. $$8^{-\frac{1}{4}}=-2$$
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