How do you rationalize a monomial denominator?

[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