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In this lesson, it will be proven that the properties of exponents are also valid for rational exponents. Moreover, these properties will be used to simplify numeric and algebraic expressions with rational exponents and roots.

Catch-Up and Review

Here are a few recommended readings before getting started with this lesson.

Product of Powers Property
Quotient of Powers Property
Power of a Power Property
Power of a Product Property
Power of a Quotient Property
Rational Exponents

Here are a few practice exercises before getting started with this lesson.

a Use the properties of exponents for integer numbers to match each algebraic expression on the left with its equivalent expression on the right.

b Write

4 2^{3}

using a rational exponent.

c Write

3^{\frac{4}{5}}

as a radical expression.

Challenge

Rewriting an Algebraic Expression

Consider the following algebraic expression.

\frac{8 x ^{3} \cdot ( x ^{\frac{1}{2}} y ) ^{5}}{3 2 x ^{2} y \cdot x ^{\frac{1}{3}}}

Can the above expression be written using only rational exponents? Using only radicals?

Discussion

Properties of Exponents

The following table shows the Properties of Exponents.

Product of Powers Property	$a^{m} a^{n} = a^{m + n}$
Quotient of Powers Property	$\frac{a ^{m}}{a ^{n}} = a^{m - n}$
Power of a Product Property	$(a b)^{n} = a^{n} b^{n}$
Power of a Quotient Property	$(\frac{a}{b})^{n} = \frac{a ^{n}}{b ^{n}}$
Power of a Power Property	$(a^{m})^{n} = a^{m \cdot n}$

These properties hold true for integer exponents. In this lesson, they will be shown to be true for rational exponents as well.

Discussion

Product of Powers Property for Rational Exponents

The multiplication of two powers with the same base

a

and rational exponents

m

and

n

results in a power with base

a

and exponent

m + n .

The expression can be undefined for some non-positive values of

a .

Therefore, this rule will only be defined for positive values of

a .

Proof

Product of Powers 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} \cdot a^{n}

is equal to

a^{m + n} .

a^{m} \cdot a^{n}

SubstituteII

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

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

▼

Simplify

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

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

MultRoot

$r a \cdot r b = r a \cdot b$

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

Recall that

p

and

q

are integer numbers. Therefore, the Product of Powers Property can be used.

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

The resulting expression can be written using only exponents.

r a^{p + q}

▼

Simplify

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

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

WriteSumFrac

Write as a sum of fractions

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

SubstituteII

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

a^{m + n}

It has been proven that

a^{m} \cdot a^{n} = a^{m + n}

for rational numbers

m

and

n .

Discussion

Quotient of Powers Property for Rational Exponents

The quotient of two powers with same base

a

and rational exponents

m

and

n

results in a power with base

a

and exponent

m - n .

The expression can be undefined for some non-positive values of

a .

Therefore, this rule will only be defined for positive values of

a .

Proof

Quotient of Powers 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 equalities,

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

can be proven to be equal to

a^{m - n} .

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

SubstituteII

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

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

▼

Simplify

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

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

DivRoot

$\frac{r a}{r b} = r \frac{a}{b}$

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

Recall that

p

and

q

are integer numbers. Therefore, the Quotient of Powers Property can be used.

r \frac{a ^{p}}{a ^{q}} = r a^{p - q}

The resulting expression can be written using only exponents.

r a^{p - q}

▼

Simplify

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

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

WriteDiffFrac

Write as a difference of fractions

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

SubstituteII

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

a^{m - n}

It has been proven that

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

for rational numbers

m

and

n .

Be aware that

a \div b

is the same as

\frac{a}{b} .

Therefore,

a^{m} \div a^{n}

is also equal to

a^{m - n},

when

a > 0

and both

m

and

n

are rational numbers.

Discussion

Power of a Product Property for Rational Exponents

Multiplying two numbers or variables and then raising the product to the power of

n,

where

n

is rational, is the same as raising the factors to the power of

n

and then multiplying them.

The expression can be undefined for some non-positive values of

a

and

b .

Therefore, this rule will only be defined for positive values of

a

and

b .

Proof

Power of a Product Property for Rational Exponents

Since

n

is a rational number, it can be written as the quotient of two integers

p

and

q,

where

q \neq = 0 .

n = \frac{p}{q}

Using the above equality,

(a b)^{n}

can be proven to be equal to

a^{n} b^{n} .

(a b)^{n}

Substitute

$n = \frac{p}{q}$

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

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

q (a b)^{p}

Recall that

p

is an integer number. Therefore, the Power of a Product Property can be used to rewrite the expression.

q (a b)^{p} = q a^{p} b^{p}

The index of the above radical is an integer number,

q .

Therefore, the obtained expression can be rewritten as the product of two radicals. Then, the definition of a rational exponent can be used.

q a^{p} b^{p}

RootProd

$q a \cdot b = q a \cdot q b$

q a^{p} q b^{p}

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

a^{\frac{p}{q}} b^{\frac{p}{q}}

Substitute

$\frac{p}{q} = n$

a^{n} b^{n}

It has been proven that

(a b)^{n} = a^{n} b^{n}

for rational

n .

Discussion

Power of a Quotient Property for Rational Exponents

Dividing two numbers or variables and then raising the quotient to the power of

n,

where

n

is rational, is the same as raising the divisor and dividend to the power of

n

and then calculating the quotient.

The expression can be undefined for some non-positive values of

a

and

b .

Therefore, this rule will only be defined for positive values of

a

and

b .

Proof

Power of a Quotient Property for Rational Exponents

Since

n

is a rational number, it can be written as the quotient of two integers

p

and

q,

where

q \neq = 0 .

n = \frac{p}{q}

Using the above equality, it can be proven that

(\frac{a}{b})^{n}

is equal to

\frac{a ^{n}}{b ^{n}} .

(\frac{a}{b})^{n}

Substitute

$n = \frac{p}{q}$

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

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

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

Recall that

p

is an integer number. Therefore, the Power of a Quotient Property can be used to rewrite the expression.

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

The index

q

of the above radical is an integer number. Therefore, the obtained expression can be rewritten as the quotient of two radicals. Then, the definition of a rational exponent can be used.

q \frac{a ^{p}}{b ^{p}}

RootQuot

$q \frac{a}{b} = \frac{q a}{q b}$

\frac{q a ^{p}}{q b ^{p}}

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

\frac{a ^{\frac{p}{q}}}{b ^{\frac{p}{q}}}

Substitute

$\frac{p}{q} = n$

\frac{a ^{n}}{b ^{n}}

It has been proven that

(\frac{a}{b})^{n} = \frac{a ^{n}}{b ^{n}}

for rational

n .

Discussion

Power of a Power Property for Rational Exponents

If a power with base

a

and rational exponent

m

is raised to the power of

n,

where

n

is rational, then the result is a power with base

a

and exponent

m \cdot n .

The expression can be undefined for some non-positive values of

a .

Therefore, this rule will only be defined for positive values of

a .

Proof

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}

SubstituteII

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

(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}

or

(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}

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Catch-Up and Review

Proof

Proof

Proof

Proof

Proof