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Chapter 0: Problem 41

Add or subtract terms whenever possible. $$3 \sqrt{18}+5 \sqrt{50}$$

Short Answer

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

Step by step solution

01

Simplify the Square Root Terms

Start by simplifying \(3 \sqrt{18}\) and \(5 \sqrt{50}\). Both 18 and 50 are not prime numbers and can be expressed as a product of perfect squares and other numbers. \(18 = 3^2 \times 2 \) and \(50 = 5^2 \times 2 \). Therefore, the square roots can be simplified as: \(3 \sqrt{18} = 3 \times 3\sqrt{2} = 9\sqrt{2}\), \(5 \sqrt{50} = 5 \times 5\sqrt{2} = 25\sqrt{2}\)
02

Add the Simplified Terms

Now add the simplified terms together: \(9\sqrt{2} + 25\sqrt{2}\). Both terms are like terms since they are both multiples of \(\sqrt{2}\). Therefore, you can add them together as you would normally do with like terms.
03

Calculate Final Result

Simply add the coefficients in front of the \(\sqrt{2}\) to obtain: \(9\sqrt{2} + 25\sqrt{2} = 34\sqrt{2}\)

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Key Concepts

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

Adding Radicals
Adding radicals might seem daunting at first, but it becomes quite simple once you understand the basics. When you need to add radicals, such as square roots, the process is similar to adding variables. You must identify and add only those radicals that have the same root value, known as being "like terms." For example, in the expression \(3 \sqrt{18}+5 \sqrt{50}\), both terms can be expressed in terms of the same root, \(\sqrt{2}\), after simplification. Once the terms become like terms, you simply add their coefficients, just as you would add regular numbers. This makes handling radicals much more manageable and straightforward.
Like Terms
In the context of adding and subtracting radicals, understanding the concept of "like terms" is vital. Like terms are terms that contain the same variable or radical. For instance, \(9\sqrt{2}\) and \(25\sqrt{2}\) are like terms because they both contain \(\sqrt{2}\). This similarity allows you to combine them by simply adding or subtracting their coefficients.
  • The coefficients are the numbers found right in front of the radicals or variables.
  • In \(9\sqrt{2}\), the coefficient is 9.
  • In \(25\sqrt{2}\), the coefficient is 25.
Adding like terms results in \(34\sqrt{2}\). Recognizing and using like terms simplifies expressions and makes calculations much easier.
Square Roots
Square roots occur frequently in algebra and can be simplified by factoring the number into a product of perfect squares and simpler numbers. For instance, \(\sqrt{18}\) can be broken down to \(\sqrt{3^2 \times 2}\), which then becomes \(3\sqrt{2}\). Similarly, \(\sqrt{50}\) can be decomposed to \(\sqrt{5^2 \times 2}\), simplifying to \(5\sqrt{2}\).
  • Look for perfect squares within the number under the square root.
  • Extract the square root of the perfect square and place it in front of the radical.
  • The remaining non-square factor stays inside the square root.
Understanding how to manipulate and simplify square roots helps to combine and simplify expressions, bringing clarity to problems involving radicals.

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Most popular questions from this chapter

Determine whether each statement makes sense or does not make sense, and explain your reasoning. The model \(P=-0.18 n+2.1\) describes the number of pay phones, \(P,\) in millions, \(n\) years after \(2000,\) so I have to solve a linear equation to determine the number of pay phones in 2010

The formula for converting Celsius temperature, \(C,\) to Fahrenheit temperature, \(F\), is $$F=\frac{9}{5} C+32$$ If Fahrenheit temperature ranges from \(41^{\circ}\) to \(50^{\circ},\) inclusive, what is the range for Celsius temperature? Use interval notation to express this range.

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. What's wrong with this argument? Suppose \(x\) and \(y\) represent two real numbers, where \(x>y .\) $$\begin{aligned}2 &>1 \\\2(y-x) &>1(y-x) \\\2 y-2 x &>y-x \\\y-2 x &>-x \\\y &>x\end{aligned}$$ This is a true statement. Multiply both sides by \(y-x\) Use the distributive property. Subtract \(y\) from both sides. Add \(2 x\) to both sides. The final inequality, \(y>x,\) is impossible because we were initially given \(x>y\)

Use the strategy for solving word problems, modeling the verbal conditions of the problem with a linear inequality. A company manufactures and sells personalized stationery. The weekly fixed cost is \(\$ 3000\) and it costs \(\$ 3.00\) to produce each package of stationery. The selling price is \(\$ 5.50\) per package. How many packages of stationery must be produced and sold each week for the company to generate a profit?

What is the discriminant and what information does it provide about a quadratic equation?
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