# Say I am asked to solve √196 by simplifying it in radical form...

#### GetThroughDiffEq

##### New member
Say I am asked to solve √196 by simplifying it in radical form...

I.E. since √196 = +/- 14

Shouldn't it be +/- 2√49 instead of just 2√49?

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#### JeffM

##### Elite Member
No.

It is true that

$$\displaystyle (-\ 14) * (-\ 14) = 196 = 14 * 14.$$

But $$\displaystyle x \in \mathbb R \text { and } x \ge 0 \implies \sqrt{x} \in \mathbb R \text { and } \sqrt{x} \ge 0.$$

So

$$\displaystyle \sqrt{196} = 14 \text { and } -\ 14 = -\ \sqrt{196}.$$

#### pka

##### Elite Member
I.E. since √14 = +/- 14
Shouldn't it be +/- 2√49 instead of just 2√49?
NO! $$\displaystyle \sqrt{14}>0$$ that is the radical is a positive number.
If $$\displaystyle x^2=81$$ then $$\displaystyle x=\pm 9$$ BUT $$\displaystyle \sqrt{81}=9$$.

#### GetThroughDiffEq

##### New member
I still don't get it. So, If I'm asked to simply √196 into radical form, what would be the answer?

2√49 or NO SOLUTION?

#### JeffM

##### Elite Member
I am also not sure what is meant by simplification in radical form. I may be mistaken, but what I would give as an answer would be to start by decomposing 196 into a product of primes and proceed.

$$\displaystyle \sqrt{196} = \sqrt{2^2 * 7^2} = \sqrt{2^2} * \sqrt{7^2} = 2 * 7 = 14.$$

It strikes me as odd to simplify

$$\displaystyle \sqrt{196} = 2\sqrt{49}$$

and stop.

#### Otis

##### Senior Member
… I am asked to solve √196 by simplifying
I don't think you were asked to "solve". I think you were asked to "simplify".

We solve equations; we simplify expressions. You were not given an equation (so there is nothing to solve); you were given an expression.

Did you see my post in your other thread, where I discussed the "principal square root"?

You were given √196, and that expression represents the principal square root of 196.

There will be no negative number, in the simplified expression, because the principal square root of any Real number is always non-negative. (That's the definition!)

Note: -14 is what you would get, were you to simplify -√196.

You can google "principal square root", to see examples and learn the concept.

#### pka

##### Elite Member
I am also not sure what is meant by simplification in radical form. I may be mistaken, but what I would give as an answer would be to start by decomposing 196 into a product of primes and proceed.
$$\displaystyle \sqrt{196} = \sqrt{2^2 * 7^2} = \sqrt{2^2} * \sqrt{7^2} = 2 * 7 = 14.$$
It strikes me as odd to simplify
$$\displaystyle \sqrt{196} = 2\sqrt{49}$$
and stop.
This question depends upon how it is written.
1. What is the value of $$\displaystyle \sqrt{196}$$? Anawer: $$\displaystyle 14$$.
2. What are the square roots of $$\displaystyle 196$$ Answer: $$\displaystyle \pm 14$$.
Notice the verb. One is plural, the other is singular.

#### Otis

##### Senior Member
… It strikes me as odd to simplify

$$\displaystyle \sqrt{196} = 2\sqrt{49}$$

and stop.
Maybe the OP didn't recognize 49 as 7×7.

If so, they wouldn't think: √49 = 7.

#### JeffM

##### Elite Member
This question depends upon how it is written.
1. What is the value of $$\displaystyle \sqrt{196}$$? Anawer: $$\displaystyle 14$$.
2. What are the square roots of $$\displaystyle 196$$ Answer: $$\displaystyle \pm 14$$.
Notice the verb. One is plural, the other is singular.
I do not see how the reply above is at all relevant to what is quoted. I was not asking about signs, an issue that I had already addressed in post # 2. I was asking about the meaning of the OP's term "simplication in radical form" and the apparent implication that the method led to

$$\displaystyle \sqrt{196} = 2\sqrt{49}.$$

#### JeffM

##### Elite Member
Maybe the OP didn't recognize 49 as 7×7.

If so, they wouldn't think: √49 = 7.

No, he clearly did recognize that 196 is the square of 14. See the second line of post # 1.

#### Otis

##### Senior Member
… [the OP] clearly did recognize that 196 is the square of 14 …
I'm not sure that's as clear as "clearly", heh. Maybe the OP got 14 by looking at the answer. Not sure …

But, I agree with your first point. Something went wrong somewhere, after stopping at 2√49.

See the second line of post # 1.
I think you mean the third line.

#### JeffM

##### Elite Member
I think you mean the third line.

In my first job (a summer job), someone teased me by saying "you can ask Jeff if you need a formula, but don't expect him to count." And it turns out to be true.