um math prep: s1.2 - non-integer exp

The natural definition for exponentiation as an abbreviation for multiplication holds for positive integers only. We've already chosen definitions for negative and zero exponents (as shown on the integer exponent page) which are consistent with our rules for exponents. For example, that a to the 0 equals 1 makes sense with the multiplication of exponents rule:
$a^n a^0 = a^{n+0} = a^n$
so, looking at the first and last entries in the equalities, a to the 0 must be one.

Rational (fractional) exponents

Fractional exponents are defined as roots, which makes sense from the rule for powers of powers:
$(a^{1/2})^2 = a^{2/2} = a^1 = a, so (a^{1/2})^2 = a$
Is it obvious that the 1/2 power of a is therefore the square root of a? Take the square root of both sides of the right-most equation above:
$(a^{1/2}) = sqrt(a)$
Another way of thinking about this is that the 1/nth power of a is the number which when raised to the n gives a--which is just the nth root of a.

Definition of roots

For positive numbers a
	$\sqrt{a}$	is the positive number whose square is a: e.g.,	$\sqrt{49} = 7$
	$\root n\of{a}$	is the positive number whose nth power is a: e.g.,	$\root 3\of{125} = 5$
For negative numbers a
	$\root n\of{a}$	is the negative number whose nth power is a if n is odd: e.g.,	$\root 3\of{-8} = -2$
	$\root n\of{a}$	is not a real number if a is negative and n is even: e.g.,	$\sqrt{-8} = ?$

Because fractional exponents are defined as roots, we have the definitions below. The rules for fractional exponents are the same as those for integer exponents. Thus we have:

Definitions:

$a^{1/n} = \root n\of a$		(a must be non-negative if n is even)
$a^{m/n} = \root n\of{a^m} = (\root n\of a)^m$		(a^m must be non-negative if n is even)

Rules for Fractional Exponents
The rules for integer exponents for multiplying, dividing and raising exponentials to powers all apply for fractional exponents. In particular, we have

$(ab)^{1/n} = a^{1/n} b^{1/n} = \root n\of a \root n\of b = \root n\of{ab}$		(the root of a product is the product of the roots)
$(a/b)^{1/n}= a^(1/n)/b^(1/n)= \root n\of a/\root n\of b= \root n\of{a/b}$		(the root of a quotient is the quotient of the roots)

Examples
As always, as you go through these be sure that you can explain each step of each example.

$27^{2/3} = (27^{1/3})^2 = 3^2 = 9$
$({M^{1/5}\over 3N^{-1/2}})^2 = {(M^{1/5})^2\over(3 N^{-1/2})^2} = {M^{2/5}\over 3^2 N^{-2/2}} = {M^{2/5}\over 9N^{-1} = {M^{2/5} N\over 9}$
${3u^2\sqrt u w\over w^{1/3}} = {3u^2 u^{1/2} w\over w^{1/3}} = 3 u^{2+1/2} w^{1-1/3} = 3 u^{5/2} w^{2/3}$

Practice

Before doing these practice problems, make sure that you can explain every step in the examples above. You should be sure that you work the practice problems until you are sure you understand them. When you get to the end of the section there are more practice problems on the section test.

Note that you can get new practice problems (for #3-5) by clicking the "Refresh" button at the bottom of the practice set.

Write the following without exponents by simplifying mentally, without using a calculator:

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More mental calculation: also write the following without exponents by simplifying mentally, without using a calculator:

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Use the rules of exponents to simplify the following expression into the form

(where p may be a rational number!)

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