How do you rationalize a monomial denominator?

[eumyang]

Why does sqrt8 simplify to sqrt4(2)? Which then simplifies to 2(sqrt2).

We use the product property of radicals:
$$\sqrt{ab} = \sqrt{a} \cdot \sqrt{b}$$
... so
$$\sqrt{8} = \sqrt{4 \cdot 2} = \sqrt{4} \cdot \sqrt{2} = 2\sqrt{2}$$

 

[lj19]

I have a question about radicals in general.
If you are adding [or subtracting] radicals or radicals that are being multiplied by a number, the radical stays the same correct? For example sqrt5+sqrt5=sqrt5? Another example is 4(sqrt5)+2(sqrt5), where the sqrt5 would stay sqrt5.
If you are multiplying [or dividing] radicals or radicals that are being multiplied by a number, the number in the radical is multiplied correct? For example, (5(sqrt10))(1(sqrt10))=5(sqrt100).
Is this correct?

 

[fzero]

I have a question about radicals in general.
If you are adding [or subtracting] radicals or radicals that are being multiplied by a number, the radical stays the same correct? For example sqrt5+sqrt5=sqrt5? Another example is 4(sqrt5)+2(sqrt5), where the sqrt5 would stay sqrt5.
If you are multiplying [or dividing] radicals or radicals that are being multiplied by a number, the number in the radical is multiplied correct? For example, (5(sqrt10))(1(sqrt10))=5(sqrt100).
Is this correct?

$$\sqrt{5}+\sqrt{5} = 2\sqrt{5}.$$

$$4\sqrt{5}+2\sqrt{5} = 6\sqrt{5}.$$

These are both examples of the distributive law for multiplication ac + bc = (a+b)c.

$$(5\sqrt{10})(1\sqrt{10}) = 5 \sqrt{100}$$

uses the associative law a(bc) = (ab)c and the commutative law ab=ba, as well as the product property of radicals.

 

[lj19]

You're correct, I forgot that there is always a 1 in front of the radical. So it's correct that when adding radicals the number in the radical does not change, but when multiplying the number in the radical is multiplied?

Could you help me understand how to solve problems with the index with radicals.
For example, [multiplying a monomial] 4(sqrt of x to the 5th power) with an index of 3 in the radical, multiplied by sqrt of 16 x to the second power with an index of 3 in the radical.
Also, how do I write this mathematically on this website?
In my notes this problem was simplified to 4(16x to the seventh power) with a 3as the index, and then simplified to 4(8x to the sixth power times 2x) with a 3 as the index.
How would I solve this problem?

 

[fzero]

You're correct, I forgot that there is always a 1 in front of the radical. So it's correct that when adding radicals the number in the radical does not change, but when multiplying the number in the radical is multiplied?

Instead of asking when the number changes, you should review the distributive law that I've been quoting and the product rule for radicals. What you say is roughly true but you can make other choices for how to express a sum or product. For example,

$$\sqrt{5} + \sqrt{5} = 2 \sqrt{5} = \sqrt{20}$$

$$(5\sqrt{10})(\sqrt{10}) = 5 \sqrt{100} = \sqrt{2500} = 50.$$

Rather than sticking to rules about whether the radical changes or not, you should learn to become comfortable with multiplication and exponents.

Could you help me understand how to solve problems with the index with radicals.
For example, [multiplying a monomial] 4(sqrt of x to the 5th power) with an index of 3 in the radical, multiplied by sqrt of 16 x to the second power with an index of 3 in the radical.

In my notes this problem was simplified to 4(16x to the seventh power) with a 3as the index, and then simplified to 4(8x to the sixth power times 2x) with a 3 as the index.
How would I solve this problem?

Do you mean

$$(4 \sqrt[3]{x^5})( \sqrt[3]{(16 x)^7}) ?$$

I would put everything under the radical and then express everything in terms of powers of 2 and x:

$$(4 \sqrt[3]{x^5})( \sqrt[3]{(16 x)^7}) = \sqrt[3]{(4^3)x^5( (2^4) x)^7} =\sqrt[3]{(2^6)(2^{28}) x^{12}} = \sqrt[3]{(2^{34})x^{12}} = 2^{11} x^4\sqrt[3]{2} .$$

I don't know if that's the same result that you got in class, but it's the simplest form for the expression, assuming that I didn't make any mistakes.

Also, how do I write this mathematically on this website?

There's some references on how to write the LaTeX code in this post:

https://www.physicsforums.com/showthread.php?t=386951

It takes some getting used to.

 

[lj19]

Thank you for the reference link. I just need to know how to know specific things for this class with radicals. In my class, the problem was simplified to 4(16x to the seventh power) then the answer was 4(8x to the sixth power times 2x) with a 3 as the index. What is an index with radicals? How do I multiply monomials like this problem?

 

[fzero]

Thank you for the reference link. I just need to know how to know specific things for this class with radicals. In my class, the problem was simplified to 4(16x to the seventh power) then the answer was 4(8x to the sixth power times 2x) with a 3 as the index. What is an index with radicals? How do I multiply monomials like this problem?

The definition of the index in the radical is

$$\sqrt[n]{a} =b$$

means that

$$b^n =a.$$

Everything you need to know specifically about multiplication of radicals is contained in the rule

$$\sqrt[n]{a} \sqrt[n]{b} = \sqrt[n]{ab}.$$

For the formulas from class, do you mean

$$\sqrt[3]{4 (16x)^7}$$

and

$$4 \sqrt[3]{2x(8x)^6} ?$$

 

[lj19]

Those are the original parts of the problem that are being multiplied.
Could you help me solve problems with multiplying monomials that have an index?
Could you explain this problem to me?
http://www.regentsprep.org/Regents/math/algtrig/ATO3/rda2.htm

 

[lj19]

Those are the original parts of the problem that are being multiplied.
Could you help me solve problems with multiplying monomials that have an index?
Could you explain this problem to me?
http://www.regentsprep.org/Regents/math/algtrig/ATO3/rda2.htm