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Why We Simplify Radicals

Reference > Mathematics > Algebra > Simplifying Radicals
 

One of the complaints high school math students often have is, "Why do we have to simplify our answers?" It certainly seems as though, if you went to all the work to get a correct answer to a complicated problem, you shouldn't be penalized for not writing it in exactly the right form!

But there is a reason to the madness of requiring mathematicians to all write their answers in the same form. It has to do with consistency. Let's suppose you have three researchers, Suzy, Mark, and Gina, and these three researchers are attempting to solve the same problem. When they're done, they want to compare their work, and make sure they're all on the right track.

Suzy comes up with the answer
48
, Mark comes up with the answer 2
12
, and Gina has 4
3
. Which one is correct?

Well, it turns out that all three values are exactly the same number - they're all approximately 6.928. But now they're stalled, because none of their answers look alike, and they have to each work things out to verify that all three of them have the same result.

The rules for simplifying radicals are designed in such a way that there is exactly one "simplest form" for a radical expression. This means that if they had followed the rules, all three of them would have given Gina's answer: 4
3
.

"Yeah," you might say, "but they could just all punch the answer into their calculators, and verify that they were the same value. That would be quicker."

Well, yes, that might be true...as long as their answers are numerical. But what if their answers are algebraic expressions? What then? What if Suzy had an answer of (x + 1)
x
, and Mark had an answer of
x3 + 2x2 + x
? Are these the same thing? And can you figure out whether they're the same by using a calculator?

You cannot.

Before you're finished with this unit, you'll be able to recognize that these are the same expression and that Suzy's is the simpler form, and you'll know how to change
x3 +2x2 + x
into its simplest form.

Questions

1.
Are
12
and 4
3
the same value?
2.
Are
24
and 2
6
the same value?
3.
Are
2
and
2
2
the same number?
4.
Are
1
1 +
2
and 1 -
2
the same number?
5.
Are
1
1 -
2
and -1 -
2
the same number?
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Simplifying Numerical Square RootsSimplifying Numerical Square Roots
 

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