Power of a Power Property for Rational Exponents

Since

m

and

n

are rational numbers, they can be written as the quotients of two integers. Let

p,

q,

and

r

be three integers such that

r \neq = 0,

m

is the quotient of

p

and

r,

and

n

is the quotient of

q

and

r .

m = \frac{p}{r} and n = \frac{q}{r}

Using the above definitions, it can be proven that

(a^{m})^{n}

is equal to

a^{m \cdot n} .

(a^{m})^{n}

$m = \frac{p}{r}$ , $n = \frac{q}{r}$

(a^{\frac{p}{r}})^{\frac{q}{r}}

$a^{\frac{m}{n}} = n a^{m}$

(r a^{p})^{\frac{q}{r}}

Recall that

a^{\frac{m}{n}}

can be defined either as

n a^{m}

(n a)^{m} .

Using the second definition, the last obtained expression can be rewritten.

(r a^{p})^{\frac{q}{r}} = (r r a^{p})^{q}

The root of a root can be expressed using only one root whose index is the product of the original indices.

(r r a^{p})^{q} \Leftrightarrow (r \cdot r a^{p})^{q}

Note that since

r

is an integer number, then

r \cdot r

is also integer. Using the fact that

n a^{m}

and

(n a)^{m}

represent the same expression

a^{\frac{m}{n}},

the power

q

can be moved inside the radical.

(r \cdot r a^{p})^{q} \Leftrightarrow r \cdot r (a^{p})^{q}

p

and

q

are integer numbers, the Power of a Power Property can be used.

r \cdot r (a^{p})^{q}

$(a^{m})^{n} = a^{m \cdot n}$

r \cdot r a^{p \cdot q}

$n a^{m} = a^{\frac{m}{n}}$

a^{\frac{p \cdot q}{r \cdot r}}

Write as a product of fractions

a^{\frac{p}{r} \cdot \frac{q}{r}}

$\frac{p}{r} = m$ , $\frac{q}{r} = n$

a^{m \cdot n}

It has been proven that

(a^{m})^{n} = a^{m \cdot n}

for rational numbers

m

and

n .