Fractional powers

Numbers: Roots

Fractional powers

Now we will look at numbers of the form #a^{\frac{m}{n}}#, where #m# and #n# are natural numbers. The exponent of #a# in this power is a fraction. Hence the following description.

Fractional powers

Let #m# and #n# be positive integers and let #a# be a non-negative number.

Instead of #\sqrt[n]{a^m}# we can write #a^{\frac{m}{n}}#.
If #a\gt0#, then, instead of #\frac{1}{\sqrt[n]{a^m}}# we also write #a^{-\frac{m}{n}}#.

These powers of #a# are called fractional powers.

We discuss why this is possible. If #p# is a third positive integer, then #\frac{m}{n} = \frac{mp}{np}# holds . If the notation of #a^{\frac{m}{n}}# for #\sqrt[n]{a^m}# is appropriate, then also #\sqrt[np]{a^{mp}}=\sqrt[n]{a^m}# has to hold. This is indeed the case, a proof of the rules of calculation for higher roots.

Rules of calculation for fractional powers

Let #a# and #b# be rational numbers and #x# and #y# positive numbers. Then the following equalities hold.

#\left(x^a\right)^b = x^{a\cdot b}#
#\left(x\cdot y\right)^a = x^a\cdot y ^a#
#x^a\cdot x^b = x^{a+b}#
#\frac{x^a}{x^b}=x^{a-b}#
#x^0 = 1#

In fact, these laws also hold if #x# and #y# are non-negative numbers and all expressions involved are defined. This means that the exponent is not allowed to be negative if the base number is equal to #0#.

Later we will see that these laws not only apply if #a# and #b# are rational, but even if they are arbitrary real numbers. The problem is not only to derive the equalities, but also to show what the meaning of #x^a# is if #a# is an arbitrary real number.

Simplify #32^{\frac{1}{5}}# into an integer.

#32^{\frac{1}{5}}=# #2#

Because, #32=2^5# so #32^{\frac{1}{5}}=\left(2^5\right)^{\frac{1}{5}}=2#.

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