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Chapter 8: Problem 101

If only like radicals can be added, why is it possible to add \(\sqrt{2}\) and \(\sqrt{8} ?\)

Short Answer

Expert verified
It is possible to add \(\sqrt{2}\) and \(\sqrt{8}\) because \(\sqrt{8}\) can be simplified to \(2 \sqrt{2}\), which makes it a like radical to \(\sqrt{2}\). The final answer of the addition is \(3 \sqrt{2}\).

Step by step solution

01

Simplify the Radicals

\(\sqrt{8}\) can be broken down into \(\sqrt{2 \times 4}\), as 2 and 4 are both factors of 8. Hence, \(\sqrt{8}\) now becomes \(2 \sqrt{2}\), because the square root of 4 equals 2.
02

Summarize the Radicals

Now, after the simplification of \(\sqrt{8}\) to \(2 \sqrt{2}\), the radical form of 2 and 8 becomes 'same', i.e., \(\sqrt{2}\). Thus, the radicands have become same for both, thereby allowing them to be added.
03

Write the Final Answer

The addition takes place to give the final answer. So, \(\sqrt{2}\ + 2 \sqrt{2}\) equals \(3 \sqrt{2}\).

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Key Concepts

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

Understanding Like Radicals
Radicals that have the same 'radicand', or number under the square root, are known as like radicals. For example, \(\sqrt{2}\) and \(2\sqrt{2}\) have the same radicand, which is 2. This similarity allows them to be added together, much like combining like terms in algebra.
To identify like radicals, focus on:
  • The number under the radical sign (square root)
  • Ensuring they are exactly the same for addition
If the radicands differ, the radicals are not alike and cannot be added directly.
Simplifying Radicals
When working with radicals, we often need to simplify them to make calculations easier. Simplifying involves expressing the radical in its simplest form. For instance, in the exercise provided, \(\sqrt{8}\) was simplified to \(2\sqrt{2}\). Here's how:
  • Break down \(8\) into factors, such as \(4 \times 2\).
  • The square root of \(4\) is \(2\), which simplifies \(\sqrt{8} = 2\sqrt{2}\).
This process allows radicals that initially seem different to become like radicals, facilitating their addition.
Adding Radicals
To add radicals, they must first be like radicals. Once you've simplified them, as discussed earlier, you can combine them just like algebraic terms.
For example, once \(\sqrt{8}\) is simplified to \(2\sqrt{2}\), you can add it to another \(\sqrt{2}\). The expression becomes:\(\sqrt{2} + 2\sqrt{2}\) which equals \(3\sqrt{2}\).
Think of adding like radicals as collecting apples of the same type; you just count them up to find the total.

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Most popular questions from this chapter

In Exercises \(53-74\), rationalize each denominator. Simplify, if possible. $$\frac{15}{\sqrt{7}+2}$$

Multiply: \(\frac{x^{2}-6 x+9}{12} \cdot \frac{3}{x^{2}-9}\) (Section 7.2, Example 3)

Make Sense? In Exercises \(90-93,\) determine whether each statement "makes sense" or "does not make sense" and explain your reasoning. I rationalized a numerical denominator and the simplified denominator still contained an irrational number.

Determine whether each statement "makes sense" or "does not make sense" and explain your reasoning. When I use the definition for \(a^{\frac{m}{n}},\) I prefer to first raise \(a\) to the \(m\) power because smaller numbers are involved.

Simplify: \(\left(2 x^{2}\right)^{-3}\). (Section \(5.7,\) Example 6 )
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