Help with exponential fractions
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Help with exponential fractions

[From: ] [author: ] [Date: 11-05-27] [Hit: ]
............
I know how to do the whole numbers such as 9^3/2...........but I can't figure out how to do one with a fraction, like this....(1/4)^3/2

I know the answer, I could use help with the steps please

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For (1/4)^(3/2), the easiest thing to do is to first to the (/2) part.

Let's rewrite it, using the property a^(bc) = (a^b)^c. So, in this case, we'll rewrite it as:

(1/4)^(3 * 1/2)
[(1/4)^(1/2)]^3

Okay, so basically, we need to calculate (1/4)^(1/2), and cube it.

Remember that ^(1/2) means square root....so what we REALLY want to do is compute the square root of 1/4. Since 1/2 * 1/2 = 1/4, the square root of 1/4 is 1/2. Does it freak your mind that the square root of a fraction is BIGGER than the fraction itself? Well, that's true for all numbers 0 < x < 1....x will be SMALLER than SQRT(x)! 1/3 is smaller than the square root of 1/3.

The other thing you could do to calculate (1/4)^(1/2) is to use the property that (a/b)^c = (a^c) / (b^c). That's a variant of (ab)^c = a^c * b^c, since division is really multiplication by a reciprocal. So, duing that gives (1^(1/2)) / (4^(1/2)). The top is 1, because 1 to the ANYTHING is 1. The bottom is the square root of 4, which is 2. So it's 1/2.

Either way, you get (1/2), and you still have to cube it, so it's 1/8.

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Prologue:

1) You know that: (2 / 3) ^ 2 --> (0.6) ^ 2 =~ 0.4 - first divide, then apply exponent

OR

is the same as: 4 / 9 =~ 0.4 - first apply the exponent, then divide

2) That 9 ^ (3/2) means:

--> 9 root square by 2 = 3
--> 3 ^ 3 = 27 (3x3x3)

So, applying the above rule:

( 1 / 4 ) ^ 3/2 --> 1^(3/2) / 4^(3/2)

--> 1 / 4^(3/2) - 1 isn't "influenced" by exponent
--> 1 / 2^3 - 4 root squared by 2 is = 2
--> 1/ 8 - applyed remaining exponent 2x2x2

So: ( 1 / 4 ) ^ 3/2 = 1/8

Are the single steps clear? hope so!

:)

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Do the numerator and denominator each by 3/2; 1 to the 3rd then sqrt is 1. 4 to the 3rd then square root is 8. so 1/8.
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