Math Is Fun Forum

careless25

Real Member

Registered: 2008-07-24

Posts: 560

Rational Exponents

I m studying for the SAT and i came upon rational exponents. I tried understanding the explanation in book but that made me more confused.
My question:
What does the number on top of the sqrt sign mean and how does it affectthe answer?

Edit: how do i make the exponent a fraction in LaTeX?

Last edited by careless25 (2008-08-06 07:18:43)

mathsyperson

Moderator

Registered: 2005-06-22

Posts: 4,900

Re: Rational Exponents

Put the exponent in curly braces after the ^, if the exponent is made of more than one character.

So, 2^2 is fine, but for something more complicated you'd need something like 2^{a+b}.
Also, you can put a \frac inside the braces to make the fraction look nicer.

To answer your first question, giving an exponent a denominator is another way of taking an nth root:

(ie. the solution of x to x^n = a)

When the exponent has a numerator other than 1, then take the number to that power before rooting:

Switching the order of powering and rooting is also allowed:

If the expression inside the brackets is an integer, then this way likely involves much simpler calculations than if you did the power first.

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careless25

Real Member

Registered: 2008-07-24

Posts: 560

Re: Rational Exponents

So what does the n do in this equation?
lets say n=3 a=5 m=2.
so it would be

What would be the answer?

Last edited by careless25 (2008-08-06 07:57:17)

mathsyperson

Moderator

Registered: 2005-06-22

Posts: 4,900

Re: Rational Exponents

Roots are basically the opposite of powers. For example, square-rooting something and then squaring it does nothing to it overall.

So the 3 above the root sign means that it's the opposite of cubing.

The answer to your question would be the same as the answer to "What cubed is 25?"
Turns out it's about 2.9ish.

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Why did the vector cross the road?
It wanted to be normal.

careless25

Real Member

Registered: 2008-07-24

Posts: 560

Re: Rational Exponents

so

and

Last edited by careless25 (2008-08-06 09:50:13)

mathsyperson

Moderator

Registered: 2005-06-22

Posts: 4,900

Re: Rational Exponents

Yup!
(It might be safer to use ≈ instead of =, but you've got the concept.)

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Why did the vector cross the road?
It wanted to be normal.

careless25

Real Member

Registered: 2008-07-24

Posts: 560

Re: Rational Exponents

1 more question!!

or

Last edited by careless25 (2008-08-06 12:30:40)

John E. Franklin

Member

Registered: 2005-08-29

Posts: 3,588

Re: Rational Exponents

25^(1/2)=5, so the 2 is the right one, but is usually just forgotten and not written.

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igloo myrtilles fourmis

Ricky

Moderator

Registered: 2005-12-04

Posts: 3,791

Re: Rational Exponents

The two isn't normally written.  The first is a bit unnecessary.

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"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."

careless25

Real Member

Registered: 2008-07-24

Posts: 560

Re: Rational Exponents

Oh! I get it!!
Question!
If u call

square root
would

be cube root?
and wat would be

?

Last edited by careless25 (2008-08-07 03:38:28)

Ricky

Moderator

Registered: 2005-12-04

Posts: 3,791

Re: Rational Exponents

Just n itself.

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"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."

careless25

Real Member

Registered: 2008-07-24

Posts: 560

Re: Rational Exponents

Thx Ricky!
how would i do a cube root on a calculator?
just show a symbol or something to look for.

mathsyperson

Moderator

Registered: 2005-06-22

Posts: 4,900

Re: Rational Exponents

Your calculator might have a button that says something like

, which would make things easy.

Otherwise, it's probably got a

button. To take the cube root of something, enter 3, then that button, then the something that you wanted cube-rooted.

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Why did the vector cross the road?
It wanted to be normal.

Ricky

Moderator

Registered: 2005-12-04

Posts: 3,791

Re: Rational Exponents

It *should* have a ^ button.  If so, you can take a cube root by pressing '^' then '0.33333' then enter.

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"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."

careless25

Real Member

Registered: 2008-07-24

Posts: 560

Re: Rational Exponents

Duh..
I should have thought of that Ricky. Thx

mathsyperson

Moderator

Registered: 2005-06-22

Posts: 4,900

Re: Rational Exponents

To be picky, it should really be "^(1/3)".
It wouldn't matter in most cases, but Ricky's way would still give slightly wrong answers.

For example, the cube root of 8 by that method gives 1.999986...

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Why did the vector cross the road?
It wanted to be normal.

Ricky

Moderator

Registered: 2005-12-04

Posts: 3,791

Re: Rational Exponents

Unless it's an infinite precision calculator, it will always give wrong answers.

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"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."