How do I solve 15^(3/8)?

Question

i need to show how I get to answer without a calculator

Answer 1

Without a calculator finding the eigth root is tricky and the answer is NOT a whole number... you can guess/check to some degree of accuracy if you know the following

Consider this...

9^(1/2) is the same as the square root of 9... The second root of 9 to the first power... which is of course 3.

so 15^(3/8) is the eigth root of 15 to the third power.... 15^3=3375..... 3375^(1/8) = 3375^(0.125) = 2.760795...

you could also resolve the parenthesis first and say that
15^(3/8) = 15^(0.375) = 2.760795....

Better to think about the equation as I described it above.... sorry i couldn't describe better without a mathmatic symbol typing tool

Answer 2

The expression can be expanded as follows:

15^(3/8) =
(15^3)^(1/8) =
3375^(1/8) =
3375^((1/2) * (1/2) * (1/2)) =
((3375^(1/2))^(1/2))^(1/2) [note the parenthesis nesting]

Now, if one uses a simple calculator, the work is easy: raising to 1/2 is the same as taking a square root, so take the square root of 3375 thrice:
sqrt(sqrt(sqrt(3375))) = 2.7607954 (approx.)

Since calculators are not allowed, you will need a manual way to calculate square roots. Newton's method (see sources below) is one of them. For calculating square roots, it is as follows (this is also called the Babylonian method, since they invented it first):

Let x_0 be an approximation of sqrt(n). Make x_1 = (x_0 + (n/x_0))/2; x_1 is nearer of sqrt(n) than x_0. Repeat for all indexes of x:

x_(j+1) = (x_j + (n/x_j))/2

Until, for some j, x_j is near enough x_(j+1) (say, 4 decimal places match).

There is another method for extracting square roots, which remembers, in form, long division. Please see the sources.

Alternatively, one can use Newton's method in the original problem, instead of in the broken down one: note that 15^(3/8) is the root of the equation

x^8 - 3375 = 0

If one knows Calculus, he can calculate the derivative of the left hand side of the equation, and use it in the general form of the Newton's method. Less hassle calculating than working through 3 square roots.

Lastly, one can use logarithms: n = 15^(3/8) is the same as log n = (3/8) * log(15). Without calculators, you would need a log table, and get a precision of, maybe, 4 decimal places.

Answer 3

15^(3 / 8) = 2.7607954

Answer 4

15^(3/8)=(15^3)^(1/8)=(((15^3)^(1/2))^(1/2))^(1/2)

=sqrt(sqrt(sqrt(15^3)))

OK?

Raise 15 to the 3rd power then take the square root of that, then take the square root of that, then take the square root of that.

Answer 5

The answer is 2.7607954

You don't need a calculator; just use Google: http://www.google.com/search?q=15%5E(3%2F8)

Answer 6

log it. here's a wiki site that can help you.
http://en.wikipedia.org/wiki/Logarithm