Fractional exponents: (7^1/2)^n = 7

[Guest]

hi, I often get hung up when I see fractional exponents and the strategy guides doesn't really cover it that much. Can anyone help break down the steps to solving this problem?

This problem is a modification from one of the OG problems.

Thanks

[P]

In this problem you would first open up the brackets by multiplying 1/2 and n.
7^1/2n = 7^1

Now you have same bases so you can rewrite this equation using only exponents
1/2n = 1

Then you solve for n by multiplying both sides by 2.
Resulting in: n=2

[RonPurewal]

hi, I often get hung up when I see fractional exponents and the strategy guides doesn't really cover it that much. Can anyone help break down the steps to solving this problem?

This problem is a modification from one of the OG problems.

Thanks

the post above mine pretty much gets it, but i want to point out the following explicitly: in most problems involving fractional exponents, it's not that terribly important that the exponent is fractional.

in this problem, for instance, the fact that the exponent is fractional doesn't actually matter at all. all that matters is the rule for taking a power of a power: multiply the exponents. (easy example: (x^3)^4 = x^12.) applying this rule to the problem you've cited gives n/2 = 1, which is plenty good to solve the equation.

--

now, if you are interested in fractional exponents, here's the dirt. just remember that you will probably NOT have to use this information; check the problem over a couple of times for easy exponent manipulations (such as the one above) before you start doing this fancy-schmancy interpretation stuff.

the numerator of a fractional exponent is a normal exponent: i.e., it raises the quantity to a power (as exponents "normally" do).
the denominator of a fractional exponent represents a root.
if the exponent is negative, then that takes a reciprocal of the quantity.
you can perform these three operations in any order you want; the order that's obviously more sensible is to take the root before raising to the power, so you don't have to deal with obscenely huge numbers.

example: (8/27)^(-2/3)
this exponent means that we have to flip, take the cube root, and square, in any order. let's use that order. (remember, all that's really important is that you take the cube root before squaring; you can do the reciprocal any time you want)
let's flip first, to get rid of the negative power: 27/8
then take the cube root: 3/2
now square: final answer = 9/4

that's how it works.

[Guest]

Thanks Ron & guest - just what I was looking for.

[rfernandez]

We're glad it was helpful.