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A power is an algebraic or numeric expression that consists of a base and an exponent. In this lesson, exponential notation will be extended to rational numbers.

Catch-Up and Review

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

Fractions
Square and Cube Roots
Powers
Power of a Power Property

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

Calculate the value of the following expressions. Write the answer as an integer number or as a fraction in its simplest form.

a

\frac{2}{7} (3)

b

144

Simplify.

c

(a)^{2}

d

3 a^{3}

e Use the Power of a Power Property to simplify the expression

(a^{3})^{3} .

Challenge

The Meaning of Exponent

The exponent of an expression indicates how many times the $base$ is multiplied by itself.

Power	Multiplication	Value
$4^{2}$	$4 \cdot 4$	$16$
$2^{3}$	$2 \cdot 2 \cdot 2$	$8$
$5^{4}$	$5 \cdot 5 \cdot 5 \cdot 5$	$625$
$1^{5}$	$1 \cdot 1 \cdot 1 \cdot 1 \cdot 1$	$1$

This is perfectly defined when the exponent is a natural number. However, what happens if an exponent is a rational number? For example, what is the value of

2^{\frac{1}{3}} ?

Discussion

Connection Between Rational Exponents and Radicals

By definition, to calculate the value of a power, the base is multiplied by itself as many times as indicated by the exponent. So, what happens with expressions where the exponent is a rational number, such as

2^{\frac{1}{3}} ?

It might be confusing to multiply

2

by itself

\frac{1}{3}

times.

2^{\frac{1}{3}} = 2 \times ?

Considering the Power of a Power Property, notice what happens when the expression

2^{\frac{1}{3}}

is raised to the third power.

(2^{\frac{1}{3}})^{3}

Simplify

PowPow

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

2^{\frac{1}{3} \cdot 3}

FracMultDenomToNumber

$\frac{a}{3} \cdot 3 = a$

2^{1}

ExponentOne

$a^{1} = a$

2

If the Power of a Power Property holds true for rational exponents, then

2^{\frac{1}{3}}

must be a number that, when raised to the third power, equals

2 .

Such a number already exists:

32 .

(32)^{3} = 2

Therefore, for this property to still work,

2

raised to the power of

\frac{1}{3}

is defined as the cube root of

2 .

2^{\frac{1}{3}} = 32

Discussion

Definition of $a^{\frac{1}{n}}$

The relationship between rational exponents and roots can be extended to any rational exponent of the form $\frac{1}{n},$ where $n$ is a natural number.

Concept

$n^{th}$ Root

For any real number

a

and natural number

n,

the expression

a^{\frac{1}{n}}

is defined as the

n^{th}

root of

a .

Note that a root with an even index is defined only for non-negative numbers. Therefore, if

n

is even, then

a

must be non-negative.

With this definition, any power with an exponent of the form

\frac{1}{n}

can be written as a radical.

Example

Rewriting a Power as a Radical

Dominika pays $62 5^{\frac{1}{4}}$ dollars per hour to study with a private tutor.

As her first homework assignment, she was given the task of writing how much she pays per hour as a radical. Write her answer without including the currency symbol.

Use the radical expression to calculate the value of

62 5^{\frac{1}{4}} .

Hint

Raising any positive number to the power of $\frac{1}{4}$ is equivalent to calculating the fourth root of the number.

Solution

To calculate the value of

62 5^{\frac{1}{4}},

rewrite it as a radical using the definition of

a^{\frac{1}{n}} .

62 5^{\frac{1}{4}}

$a^{\frac{1}{4}} = 4 a$

4625

WritePow

Write as a power

4 5^{4}

CalcRoot

Calculate root

5

Pop Quiz

Calculating Powers Using Radicals

Calculate the values of the given powers with rational exponents by using a radical.

The applet that generates powers with rational exponents

Example

Finding a Radical and Rewriting It as a Power

YBC

7289

is an ancient Babylonian clay tablet believed to be the work of a student who lived in southern Mesopotamia around the year

1700 BC .

The tablet contains an extremely accurate approximation of the length of the diagonal of a square with side length

1 .

The length of the diagonal is given to an accuracy of six decimal digits, the greatest known computational accuracy in the ancient world. Using the Pythagorean Theorem, calculate the exact length of the diagonal. Give the answer as a radical.

Write the length of the diagonal using a rational exponent.

Hint

Consider a right triangle in which the legs are two sides of the square and the hypotenuse is the diagonal.

Solution

Consider the right triangle formed by two sides of the square and its diagonal.

Here, the length of both legs is

1 .

Therefore, by substituting

a = 1

and

b = 1

into the Pythagorean Theorem, the length of the diagonal

c

can be found.

a^{2} + b^{2} = c^{2}

SubstituteII

$a = 1$ , $b = 1$

1^{2} + 1^{2} = c^{2}

Solve for

c

BaseOne

$1^{a} = 1$

1 + 1 = c^{2}

AddTerms

Add terms

2 = c^{2}

SqrtEqn

$LHS = RHS$

2 = c^{2}

SqrtPowToNumber

$a^{2} = a$

2 = c

RearrangeEqn

Rearrange equation

c = 2

The length of the diagonal, written as a radical, is exactly

2

units. This number can be also expressed as a power with a rational exponent of

\frac{1}{2} .

2

SqrtToPowD

$a = a^{\frac{1}{2}}$

2^{\frac{1}{2}}

Discussion

Definition of a Rational Exponent

Note that the numerator of a rational number does not have to be $1,$ but could also be any integer number. Therefore, rational numbers with a numerator different than $1$ should also be considered as rational exponents.

Concept

Rational Exponent

When a number is raised to the power of a fraction, that fraction is the number's rational exponent. Such an expression is equivalent to a root.

Notice that the denominator of the rational exponent gives the index of the root, while the numerator gives the power to which

a

is raised. Since

n

is a denominator, it cannot be zero. Moreover, if

n

is an even number, then

a^{m}

must be non-negative.

Discussion

Another Radical Equivalent to $a^{\frac{m}{n}}$

Using the Power of a Power Property, another expression equivalent to

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

involving a radical can be found.

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

Simplify

MoveNumRight

$\frac{a}{b} = \frac{1}{b} \cdot a$

a^{\frac{1}{n} \cdot m}

ProdInExponent

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

(a^{\frac{1}{n}})^{m}

PowToRootD

$a^{\frac{1}{n}} = n a$

(n a)^{m}

Therefore,

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

can also be defined as

(n a)^{m} .

Note that if

n

is an even number, then

a

must be non-negative. In conclusion, there are two definitions for

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

a^{\frac{m}{n}} = n a^{m} or a^{\frac{m}{n}} = (n a)^{m}

Example

Expressing a Rational Exponent as a Radical

Ignacio ate some cake at a birthday party. When he arrived home, he told his parents that he had $8^{\frac{2}{3}}$ slices.

Ignacio telling his parents how many slices of the cake he ate

His mother, who is a math teacher, asked him to express the number of slices that he ate as a radical.

His father, who does not understand powers at all, asked him to calculate the value of this exponential expression.

Hint

Raising any number to the power of $\frac{2}{3}$ is equivalent to calculating the cube root of the square of the number.

Solution

To calculate the value of

8^{\frac{2}{3}},

it should be first rewritten as a radical. In order to do that, the definition of a rational exponent can be used.

8^{\frac{2}{3}}

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

3 8^{2}

CalcPow

Calculate power

364

Now Ignacio is able to give an answer for his mother. He can say that he ate

364

slices of the cake. To answer his father, Ignacio needs to find the value of the found radical.

364

Rewrite

Rewrite $64$ as $4^{3}$

3 4^{3}

RootPowToNumber

$3 a^{3} = a$

4

Discussion

Principal Root

Now, focus on roots with an even index. For example, consider

3 .

By definition,

3

is a number that, when raised to the second power, equals

3 .

(3)^{2} = 3

What about all the numbers that are equal to

3

when raised to the second power? There are two such numbers,

3

and

- 3 .

x^{2} = 3 \Rightarrow x = 3 or x = - 3

To avoid complicating the definitions of

a^{\frac{1}{n}}

and

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

positive

3

is conventionally defined as the principal root. Therefore, for any even number

n,

n a

is defined as the positive number that, when raised to the

n th

power, equals

a .

Closure

Translating Between Radicals and Rational Exponents

It is important to know how to write expressions with rational exponents as radicals. Sometimes it is required to simplify an expression by using only radicals. Consider the following example.

Simplify the expression and rewrite it using radicals only . 3^{\frac{3}{5}} \cdot 2^{\frac{1}{5}}

Hint

Start by writing both numeric expressions using radicals.

Solution

Start by writing both factors in the expression using radicals.

3^{\frac{3}{5}} \cdot 2^{\frac{1}{5}}

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

5 3^{3} \cdot 2^{\frac{1}{5}}

PowToRootD

$a^{\frac{1}{n}} = n a$

5 3^{3} \cdot 52

MultRoot

$5 a \cdot 5 b = 5 a \cdot b$

5 3^{3} \cdot 2

CalcPow

Calculate power

5 27 \cdot 2

Multiply

554

Moreover, it is important to know how to write radical expressions using rational exponents, since rational and integer exponents have the same properties. Simplifying expressions involving exponents may be much easier than simplifying expressions involving radicals. Consider one last example.

Simplify the expression and rewrite it using exponents only . 2^{\frac{1}{3}} \cdot 2

Hint

Write $2$ using a rational exponent and then use the Product of Powers Property.

Solution

The expression can be simplified by writing

2

using a rational exponent and then using the Product of Powers Property.

2^{\frac{1}{3}} \cdot 2

SqrtToPowD

$a = a^{\frac{1}{2}}$

2^{\frac{1}{3}} \cdot 2^{\frac{1}{2}}

MultPow

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

2^{\frac{1}{3} + \frac{1}{2}}

ExpandFrac

$\frac{a}{b} = \frac{a \cdot 2}{b \cdot 2}$

2^{\frac{2}{6} + \frac{1}{2}}

ExpandFrac

$\frac{a}{b} = \frac{a \cdot 3}{b \cdot 3}$

2^{\frac{2}{6} + \frac{3}{6}}

AddFrac

Add fractions

2^{\frac{5}{6}}

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Catch-Up and Review

Challenge

The Meaning of Exponent

Discussion

Connection Between Rational Exponents and Radicals

Discussion

Definition of $a^{\frac{1}{n}}$

Concept

$n^{th}$ Root

Example

Rewriting a Power as a Radical

Hint

Solution

Pop Quiz

Calculating Powers Using Radicals

Example

Finding a Radical and Rewriting It as a Power

Hint

Solution

Discussion

Definition of a Rational Exponent

Concept

Rational Exponent

Discussion

Another Radical Equivalent to $a^{\frac{m}{n}}$

Example

Expressing a Rational Exponent as a Radical

Hint

Solution

Discussion

Principal Root

Closure

Translating Between Radicals and Rational Exponents

Hint

Solution

Hint

Solution