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Chapter 0: Problem 122

Explain how to add \(\sqrt{3}+\sqrt{12}\)

Short Answer

Expert verified
The sum of \(\sqrt{3} + \sqrt{12}\) simplifies to \(3 \sqrt{3}\).

Step by step solution

01

Simplify \(\sqrt{12}\)

Using the prime factorization method, 12 can be broken down into 2*2*3. Hence, the square root of 12, following the product property of square roots, becomes \(\sqrt{2^2 * 3} = 2 \sqrt{3}\).
02

Add the simplified square roots

Now, add \(\sqrt{3}\) and the simplified version of \(\sqrt{12}\), which is \(2\sqrt{3}\), to get \(\sqrt{3} + 2 \sqrt{3}\). By combining like terms, this equals \(3 \sqrt{3}\).

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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 a method used to break down a number into its basic building blocks, called prime numbers. A prime number is a number greater than 1 that has no divisors other than 1 and itself. To perform prime factorization, you continually divide the number by its smallest prime factor until you're left with 1.

For example, for the number 12 mentioned in the exercise, we start by dividing by the smallest prime number, which is 2:
  • 12 梅 2 = 6
  • 6 梅 2 = 3
Now, 3 is also a prime number, and you can't divide it further by 2, so we're done.

This means 12 can be expressed as a product of prime numbers: 2 * 2 * 3. This step is crucial in simplifying square roots, as we'll see in the next section.
Product Property of Square Roots
The product property of square roots states that the square root of a product is equal to the product of the square roots of the factors. In mathematical terms, it means \[\sqrt{a \cdot b} = \sqrt{a} \cdot \sqrt{b}\]

In our exercise, after finding the prime factorization of 12 as 2 * 2 * 3, we want to simplify \(\sqrt{12}\).

You apply the product property of square roots, \(\sqrt{12} = \sqrt{2^2 \cdot 3}\). Then, it becomes \(\sqrt{2^2} \cdot \sqrt{3} = 2 \cdot \sqrt{3}\). This means \(\sqrt{12}\) simplifies to \(2\sqrt{3}\).

Understanding this property helps you work with square roots more flexibly and recognize patterns which assist in simplification.
Combining Like Terms
Combining like terms allows you to simplify expressions by adding together terms that have the same variable or radical part. In the context of square roots, like terms are those with the same radicand, which is the number under the square root symbol.

In the exercise, you have \(\sqrt{3} + 2\sqrt{3}\). Both terms have \(\sqrt{3}\) as their radical part, allowing you to combine them as follows:
  • The coefficients (numbers in front of the square roots) are 1 and 2, respectively.
  • Add these coefficients: 1 + 2 = 3.
Thus, the expression simplifies to \(3\sqrt{3}\).

Combining like terms makes expressions easier to work with and is a fundamental technique in algebra. It not only simplifies your calculations but also makes it easier to see relationships between different parts of a mathematical expression.

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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. I evaluated \(\frac{3 x-3}{4 x(x-1)}\) for \(x=1\) and obtained 0.

Use the strategy for solving word problems, modeling the verbal conditions of the problem with a linear inequality. On two examinations, you have grades of 86 and \(88 .\) There is an optional final examination, which counts as one grade. You decide to take the final in order to get a course grade of A, meaning a final average of at least 90 . a. What must you get on the final to earn an A in the course? b. By taking the final, if you do poorly, you might risk the B that you have in the course based on the first two exam grades. If your final average is less than \(80,\) you will lose your \(\mathrm{B}\) in the course. Describe the grades on the final that will cause this to happen.

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. $$\frac{x^{2}-25}{x-5}=x-5$$

What is a compound inequality and how is it solved?

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