SOLUTION: Solve. (Write answers using no spaces. For questions with two solutions, place a comma between the answers) (x+1)^((3/2))-2=25

Algebra ->  Exponential-and-logarithmic-functions -> SOLUTION: Solve. (Write answers using no spaces. For questions with two solutions, place a comma between the answers) (x+1)^((3/2))-2=25      Log On


   

Question 595690: Solve. (Write answers using no spaces. For questions with two solutions, place a comma between the answers) (x+1)^((3/2))-2=25
Answer by jsmallt9(3758) About Me  (Show Source):
You can put this solution on YOUR website!
%28x%2B1%29%5E%28%283%2F2%29%29-2=25
With equations like this , where the variable is in an expression being raised to a power, you want to start by isolating the expression and its exponent. This means the 2 has to go. Adding 2 to each side we get:
%28x%2B1%29%5E%28%283%2F2%29%29=27

Next you want the exponent to "disappear". More correctly, you want the exponent to turn into a 1. With a fractional exponent some people like to do this in two steps. But not me!? To turn the exponent into a 1
  • We will be raising both sides of the equation by some power. We just have to figure out the right power.
  • When we raise the left side to a power we will have a power of a power. The exponent rule for this is to multiply exponents. So we have to ask ourselves, "3/2 times what is equal to 1?"
  • Since multiplying reciprocals always results in a 1, we need to raise each side to the reciprocal of 3/2 power. The reciprocal of 3/2 is 2/3.
In summary, we can turn the exponent into a one by raising both sides of the equation be the reciprocal of the exponent:
%28%28x%2B1%29%5E%28%283%2F2%29%29%29%5E%282%2F3%29=%2827%29%5E%28%282%2F3%29%29
After simplifying the left side we have:
x%2B1+=+27%5E%28%282%2F3%29%29

Now we just need to figure out what the right side is. If you have trouble with negative and/or fractional exponents I find that factoring the exponent in a certain way can help:
  1. If the exponent is negative, factor out a -1.
  2. If the exponent is fractional and the numerator is not a 1, factor out the numerator.
Factoring the exponent of 2/3 this way we get:
x%2B1+=+27%5E%28%282%2A%281%2F3%29%29%29
With the exponent factored in this way, each factor tells us an operation to perform:
  • If there is a factor of -1, a reciprocal must be found.
  • If there is a factor that is a fraction (which should have a numerator of 1), then a root will be found. (1/2 means square root, 1/3 means cube root, 1/4 means 4th root, etc.)
  • A whole number factor means what it usually means (2 means square, 3 means cube etc.)
Your exponent has factors of 2 and 1/3. The 2 tells us to square and the 1/3 tells us to find a cube root.

The order in which we do these operations makes no difference! We can square first and then find the cube root or vice versa. So pick the order that looks easiest. Since 27 happens to be a perfect cube, 3%5E3+=+27, I'm going to do the cube root first:
x%2B1+=+3%5E%282%29
Since the cube root has been done, I removed the 1/3 from the exponent. And 3 squared is easy (much easier than squaring 27):
x + 1 = 9

Now that the right side has been simplified, we just have to solve for x. Subtracting 1 we get:
x = 8

Last of all we check our answer. (Checking is not optional when solving equations like this.) Use the original equation to check:
%28x%2B1%29%5E%28%283%2F2%29%29-2=25
Checking x = 8:
%28%288%29%2B1%29%5E%28%283%2F2%29%29-2=25
Simplifying:
%289%29%5E%28%283%2F2%29%29-2=25
Using what we learned earlier about fractional exponents, the 3/2 tells us to cube and find a square root (in either order). The square root of 9 is 3 and 3 cubed is 27 so now we have:
27 - 2 = 25
25 - 25 Check!

Note: If x = 8 had not checked then we would have rejected it. This would have left us with no solution, meaning that the original equation was impossible to begin with.