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Exercises: Perform the operations indicated

2√18 + √32

ManuelBautista2019
Feb 15, 2018

#1**+2 **

In order to add these radicals, the radicals must be placed in simplest radical form. When a radical is in this form, the radicand (the number or expression contained inside the radical) must not have any perfect square factors, excluding 1. This process changes the way the radical looks, but it does not actually change the value.

Putting the radicals in simplest form requires one to identify a perfect square factor of the radicand. Let's try the first radical, \(2\sqrt{18}\). In this case, the radicand is 18. A perfect square factor of 18 is 9 since 9*2=18. Let's break up the radicand in this fashion.

\(2\sqrt{18}\Rightarrow 2\sqrt{9*2}\)

It is probably obviousn that both expressions are equivalent in value. Because a multiplication operation is being performed inside the radical and the multiplicand and multiplier are both positive, it is possible to split up the radical into two separate parts.

\(2\sqrt{9*2}\Rightarrow 2\sqrt{9}*\sqrt{2}\)

This manipulation of the original term has now created two radicals: \(\sqrt{9}\text{ and }\sqrt{2}\). Notice that \(\sqrt{9}\) can be simplified into something rational. In other words, \(\sqrt{9}=3\). We can use this to simplify the radical even further.

\(2\textcolor{red}{\sqrt{9}}*\sqrt{2}=2*\textcolor{red}{3}*\sqrt{2}=6\sqrt{2}\)

Notice that the square root of 2 remains. The radicand, 2, has no perfect square factors, other than 1. I explained in the beginning that this indicates that a particular radical is in simplest form. In fact, we can use this exact process to simplify \(\sqrt{32}\). I will use the same process, but I will skip the explanation portion.

\(\sqrt{32}=\sqrt{4*8}=\sqrt{4}\sqrt{8}=2\sqrt{8}\)

Actually, I need to explain something else for you! Ideally, you want to identify the LARGEST perfect square factor. In the preceding work I showed, I identified that 4 is a perfect square factor of the radicand, 32. This is true, but 4 is not the largest perfect square factor. If you keep searching, you will quickly realize that 16 is the LARGEST perfect square factor of 32.

\(\sqrt{32}=\sqrt{16*2}=\sqrt{16}\sqrt{2}=4\sqrt{2}\)

Now,\(\sqrt{32}\) is in simplest radical form. Sometimes, though, identifying the LARGEST perfect square factor may be impractical--especially if you are dealing with enormous numbers. Let's go back to the original approach I showed:

\(\sqrt{32}=\sqrt{4*8}=\sqrt{4}\sqrt{8}=2\sqrt{8}\)

Now, the radicand is 8. It, too, has a perfect square factor, which is 4. This means you can simplify further.

\(2\sqrt{8}=2*\sqrt{4*2}=2\sqrt{4}*\sqrt{2}=2*2\sqrt{2}=4\sqrt{2}\)

Look at that! We arrived at the same answer as we did before. Therefore, always be sure that the resulting radical does not have any perfect square factors. Let's get back to the original problem.

\(2\sqrt{18}+\sqrt{32}\\ 6\sqrt{2}+4\sqrt{2}\)

The whole point of putting both radicals into simplest form is that now they have identical radicals. We can now them together.

\(6\sqrt{2}+4\sqrt{2}=10\sqrt{2}\)

I know that this is a long-winded explanation for such a simple problem. If you are confused, just ask away! I'll be glad to answer.

TheXSquaredFactor
Feb 15, 2018

#1**+2 **

Best Answer

In order to add these radicals, the radicals must be placed in simplest radical form. When a radical is in this form, the radicand (the number or expression contained inside the radical) must not have any perfect square factors, excluding 1. This process changes the way the radical looks, but it does not actually change the value.

Putting the radicals in simplest form requires one to identify a perfect square factor of the radicand. Let's try the first radical, \(2\sqrt{18}\). In this case, the radicand is 18. A perfect square factor of 18 is 9 since 9*2=18. Let's break up the radicand in this fashion.

\(2\sqrt{18}\Rightarrow 2\sqrt{9*2}\)

It is probably obviousn that both expressions are equivalent in value. Because a multiplication operation is being performed inside the radical and the multiplicand and multiplier are both positive, it is possible to split up the radical into two separate parts.

\(2\sqrt{9*2}\Rightarrow 2\sqrt{9}*\sqrt{2}\)

This manipulation of the original term has now created two radicals: \(\sqrt{9}\text{ and }\sqrt{2}\). Notice that \(\sqrt{9}\) can be simplified into something rational. In other words, \(\sqrt{9}=3\). We can use this to simplify the radical even further.

\(2\textcolor{red}{\sqrt{9}}*\sqrt{2}=2*\textcolor{red}{3}*\sqrt{2}=6\sqrt{2}\)

Notice that the square root of 2 remains. The radicand, 2, has no perfect square factors, other than 1. I explained in the beginning that this indicates that a particular radical is in simplest form. In fact, we can use this exact process to simplify \(\sqrt{32}\). I will use the same process, but I will skip the explanation portion.

\(\sqrt{32}=\sqrt{4*8}=\sqrt{4}\sqrt{8}=2\sqrt{8}\)

Actually, I need to explain something else for you! Ideally, you want to identify the LARGEST perfect square factor. In the preceding work I showed, I identified that 4 is a perfect square factor of the radicand, 32. This is true, but 4 is not the largest perfect square factor. If you keep searching, you will quickly realize that 16 is the LARGEST perfect square factor of 32.

\(\sqrt{32}=\sqrt{16*2}=\sqrt{16}\sqrt{2}=4\sqrt{2}\)

Now,\(\sqrt{32}\) is in simplest radical form. Sometimes, though, identifying the LARGEST perfect square factor may be impractical--especially if you are dealing with enormous numbers. Let's go back to the original approach I showed:

\(\sqrt{32}=\sqrt{4*8}=\sqrt{4}\sqrt{8}=2\sqrt{8}\)

Now, the radicand is 8. It, too, has a perfect square factor, which is 4. This means you can simplify further.

\(2\sqrt{8}=2*\sqrt{4*2}=2\sqrt{4}*\sqrt{2}=2*2\sqrt{2}=4\sqrt{2}\)

Look at that! We arrived at the same answer as we did before. Therefore, always be sure that the resulting radical does not have any perfect square factors. Let's get back to the original problem.

\(2\sqrt{18}+\sqrt{32}\\ 6\sqrt{2}+4\sqrt{2}\)

The whole point of putting both radicals into simplest form is that now they have identical radicals. We can now them together.

\(6\sqrt{2}+4\sqrt{2}=10\sqrt{2}\)

I know that this is a long-winded explanation for such a simple problem. If you are confused, just ask away! I'll be glad to answer.

TheXSquaredFactor
Feb 15, 2018