SOLUTION: x^-3/2=1/729

Algebra ->  Exponents-negative-and-fractional -> SOLUTION: x^-3/2=1/729      Log On


   

Question 398616: x^-3/2=1/729
Answer by jsmallt9(3758) About Me  (Show Source):
You can put this solution on YOUR website!
x%5E%28-3%2F2%29=1%2F729
The solution to this problem will be of the form
x = something
And since x and x%5E1 are the same thing, one way to look at the task ahead is: How do we change an exponent of -3/2 into a 1? To accomplish this we will put together 3 ideas:
  • It is OK to raise both sides of an equation to the same power.
  • The rule for exponents when raising a power to a power is to multiply the exponents.
  • Multiplying reciprocals always results in 1!

So by raising both sides of this equation to the reciprocal of -3/2 power, the exponent will turn into a 1. The reciprocal of -3/2 is -2/3:
%28x%5E%28-3%2F2%29%29%5E%28-2%2F3%29=%281%2F729%29%5E%28-2%2F3%29
On the left we end up with x, as planned:
x+=+%281%2F729%29%5E%28-2%2F3%29
Now we just need to simplify the right side. If you have trouble with negative and/or fractional exponents,I find that it can be helpful to factor the exponent in a certain way:
  1. If the exponent is negative, factor out a -1.
  2. If the exponent is fractional and its numerator is not a 1, factor out the numerator.

Factoring the exponent of -2/3 in this way we first factor out -1:
x+=+%281%2F729%29%5E%28-1%2A%282%2F3%29%29
The fraction in the exponent does not have a numerator of 1 so we factor out the 2:
x+=+%281%2F729%29%5E%28-1%2A2%2A%281%2F3%29%29
Looking at the factors in the exponent we can see that there will be three operations to perform:
  • The -1 (as an exponent) tells us that we will find a reciprocal.
  • The 2 (as an exponent) tells us that we willbe squaring.
  • The 1/3 (as an exponent) tells us (if you remember what fractional exponents mean) that we will be finding a cube root.

And since multiplication is Commutative, we can do these three operations in any order we choose! So let's choose the order that looks easiest. Finding a reciprocal of a fraction is pretty simple (just flip it upside down). And since the 1 in the fraction will end up in the denominator the reciprocal of 1/729 ends up being the nice whole number 729! So it looks like starting with the reciprocal is the easiest first step. Next we will square or find a cube root. Since squaring 729 and then finding a cube root does not look appealing, I am going to find a cube root and then square. Rearranging the factors in the chosen order we get:
x+=+%281%2F729%29%5E%28-1%2A%281%2F3%29%2A2%29
which is equal to:
x+=+%28%28%281%2F729%29%5E%28-1%29%29%5E%281%2F3%29%29%5E2
Now we can simplify, from the inside out:
x+=+%28%28729%2F1%29%5E%281%2F3%29%29%5E2
or
x+=+%28%28729%29%5E%281%2F3%29%29%5E2
With a little effort we can find that the cube root of 729 is 9:
x+=+%289%29%5E2
which simplifies to:
x = 81

Note: If you choose a different order, the answer works out the same!