Using the product rule to simplify √(45x^2) (don't understand the process or why absolute value is used)

[GetThroughDiffEq]

Hello everyone,

I'm working through Blitzer's Precalculus essentials 4e.

[This is for personally understanding only and not homework. I want to understand as much math as possible before starting a new degree program (Biology at the min. to as far as biomedical engineering].

As you can see, I'm stuck on the product rule.

Here are the questions I am having trouble with:

USE THE PRODUCT RULE TO SIMPLIFY:

15. √(45x^2)

My answer: 3√(5x^2)

Correct Answer: 3 |x| √5

Notes: I have no idea why I need to use an absolute value!

21. √(2x^2) * (√6x)

My answer: 1√(x^2)

Correct Answer: 2x(√3x)

Notes: Why don't we take each square root before multiplying?

Thanks in advance.

 

[Dr.Peterson]

Your are referring to the "product rule for radicals". How does your book state this rule?

The rule that you are having most trouble with is the rule (which the book has probably also stated and explained) [MATH]\sqrt{a^2} = |a|[/MATH]. The reason the rule is not [MATH]\sqrt{a^2} = a[/MATH] is that the latter would be false for an example like a = -3: [MATH]\sqrt{a^2} = \sqrt{(-3)^2} = \sqrt{9} = 3 \ne a[/MATH]. (Many books, after teaching this, will state in most problems later on that you are to assume variables are positive, so that you don't need to use absolute values.)

In exercise 15, you just didn't apply any such rule; is there a reason you stopped where you did?

In exercise 21, I have no idea what you did; can you explain your steps?

The reason you are expected to combine the two roots into one is just that it won't be fully simplified until you do. You could simplify each radical separately first; that just wastes effort sometimes.

 

[topsquark]

USE THE PRODUCT RULE TO SIMPLIFY:

15. √45x^2

It appears that for your first problem you have to simplify [math]\sqrt{45 x^2}[/math]? If you don't know how to code LaTeX or anything, please at least use parenthesis to make the line easier to interpret. For example: √45x^2 should be √(45x^2).

-Dan

 

[GetThroughDiffEq]

I just redid number 15:

√(15*3x^2)
√(353x^2)
3√(5*3x^2)

Where I'm stuck is how does 3x^2 simplify to |x|? It makes no sense to me.

Before, I got 3√(5x^2) because I followed this method that got correct answers for the radicals:

My thinking was: 3^2 = 9, therefore 9*(5x^2)=√(45x^2).

 

[topsquark]

Tiny steps:
[math]\sqrt{45x^2} = \sqrt{45} \cdot \sqrt{x^2} = \sqrt{9 \cdot 5} \cdot \sqrt{x^2} = \sqrt{9} \cdot \sqrt{5} \cdot \sqrt{x^2}[/math]
[math]= 3 \cdot \sqrt{5} |x|[/math]The [math]\sqrt{x^2} = |x| [/math] ala Dr.Peterson.

-Dan

 

[Steven G]

I guess you did not understand what Dr Peterson was saying so I will give it a try.

Consider these problems:
[math]\sqrt{3^2} = \sqrt{9}=3[/math][math]\sqrt{5^2} = \sqrt{25}=5[/math][math]\sqrt{(-4)^2} = \sqrt{16}=4[/math][math]\sqrt{(-7)^2} = \sqrt{49}=7[/math]Can you think of another function where you
input 3 and get 3 back
input 5 and get 5 back
input -4 and get 4 back and
input -7 and get 7 back?

The answer is the absolute value function.
Note | 3 | =3
| 5 | =5
| -4 | =4
| -7 | =7

This makes you think that [math]\sqrt{x^2} = |x|[/math]

 

[Otis]

… SIMPLIFY … √(45x^2) …

… Answer: 3 |x| √5 … I have no idea why I need to use an absolute value …

Have you learned about "principal square root"?

Every positive number has two square roots: one positive and one negative. The positive square root is called the principal square root.

Dr. Peterson used 9 as an example. The number 9 has two square roots: 3 and -3. The principal square root is 3 (the positive one).

When you're given a square root expression (like √9), it always represents the principal square root.

So the expression √9 equals 3.

If you wanted to express the negative square root (which is -3), then you'd need to write -√9.

Now let's look at the same situation, where the square is unknown: x^2

If you're given the expression √(x^2), then it represents the principal square root (the positive one). Since we don't know the actual value of x, we can't be sure that symbol x represents a positive number (because x could be the negative root of x^2). To ensure that we write down the principal square root (the positive one), we must enclose x in absolute-value symbols.

√(x^2) = |x|

This always works, whether x itself is positive, negative or zero. (Try it, using some Real values for x.)

Dr. Peterson also mentioned the possibility that textbooks or math courses sometimes state up front that all variables represent non-negative numbers. Be mindful of such disclaimers. For example, in Blitzer's Introductory Algebra for College Students, he shows the following simplification exercise with answer.

\(\displaystyle \sqrt{\frac{x^2}{36}} = \frac{x}{6}\)

In that example, Blitzer doesn't write \(\frac{|x|}{6}\). Why not? Because he had already declared that variables in that chapter always represent a positive number (that disclaimer appears on the first page of the chapter, so we must pay careful attention to materials).

Other times, you might see the following convention (instead of absolute-value symbols):

√(x^2) = x, where x ≥ 0

These are all different ways to express the fact that symbol √ means the principal square root (a non-negative number).

Use -√ to express the negative root.

… √(2x^2) * (√6x) …

… Why don't we take each square root before multiplying?

Because √2 and √6 are both irrational numbers, so there's no way to write the exact values of those square roots (as pure numbers).

√2 is roughly 1.414213562373095, but that's not exact.

√6 is roughly 2.449489742783178, but that's also an approximation.

Simplifying algebraic expressions (like the two in your exercise) is not about evaluating decimal approximations. Simplifying is all about finding an exact expression that has a simpler form that the one you started with.

?