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When solving geometry problems, radical expressions — like square roots or cube roots — are present in a lot of concepts and formulas. For example, the diagonal of a square is calculated by

a 2,

where

a

is the side length of the square. This lesson will explore the operations on more general radical expressions and how radicals might apply to geometry and physics.

Catch-Up and Review

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

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

a Write

8^{\frac{2}{5}}

as a radical.

b Consider the following algebraic expression. Assume that the variable

x

is positive.

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

What is the simplest form of the expression?

c Write the following expression in its simplest form.

\frac{1 + 1 6}{9 - 2 2}

Challenge

Length of a Triangular Fencing

Ignacio is planning to build an astronomical observatory in his garden. The building will be enclosed by a fence with a triangular shape. The dimensions of Ignacio's garden are presented in the following diagram.

In this diagram, all dimensions are measured in meters. Although some side lengths are still not decided, help Ignacio calculate the length of the fence $L (x)$ with respect to $x .$ What is the value of $L (5) ?$

Discussion

$n^{th}$ Root

The

n^{th}

root of a real number

a

expresses another real number that, when multiplied by itself

n

times, will result in

a .

In addition to the radical symbol, the notation is made up of the radicand

a

and the index

n .

The resulting number is commonly called a radical. For example, the radical expression

416

is the fourth root of

16 .

Notice that

416

simplifies to

2

because

2

multiplied by itself

4

times equals

16 .

416 = 4 2^{4} = 2

The general expression

n a

represents a number which equals

a

when multiplied by itself

n

times.

$n times n a \cdot n a \cdot \dots \cdot n a = a or (n a)^{n} = a$

Some numbers have more than one real $n^{th}$ root. For example, $16$ has two fourth roots, $2$ and $- 2,$ because both $2^{4}$ and $(- 2)^{4}$ are equal to $16 .$ The number of real $n^{th}$ roots depends on the sign of a radicand $a$ and an integer $n .$

	$n$ is Even	$n$ is Odd
$a > 0$	Two unique real $n^{th}$ roots, $n a$ and $- n a$	One real $n^{th}$ root, $n a$
$a = 0$	One real $n^{th}$ root, $n 0 = 0$	One real $n^{th}$ root, $n 0 = 0$
$a < 0$	No real $n^{th}$ roots	One real $n^{th}$ root, $n a$

By the definition of an $n^{th}$ root, calculating the $n^{th}$ power of the $n^{th}$ root of a number $a$ results in the same number $a .$ The following formula shows what happens if these two operations are swapped.

Rule

$n^{th}$ Roots of $n^{th}$ Powers

To simplify an $n^{th}$ root, the radicand must first be expressed as a power. If the index of the radical and the power of the radicand are equal such that $n a^{n},$ the radical expression can be simplified as follows.

n a^{n} = {∣ a, ∣ a ∣, if n is odd if n is even

The absolute value of a number is always non-negative, so when $n$ is even, the result will always be non-negative. Consider a few example $n^{th}$ roots that can be simplified by using the formula.

$a$	$n$	$n a^{n}$	Is $n$ even?	Simplify
$4$	$3$	$3 4^{3}$	$\times$	$4$
$- 6$	$5$	$5 (- 6)^{5}$	$\times$	$- 6$
$2$	$6$	$6 2^{6}$	$✓$	$∣ 2 ∣ = 2$
$- 3$	$4$	$4 (- 3)^{4}$	$✓$	$∣ - 3 ∣ = 3$

Proof

Informal Justification

To write the expression for $n a^{n},$ there are two cases to consider.

$a \geq 0$
$a < 0$

Both cases will be considered one at a time.

$a \geq 0$

Start by noting that since

a

is non-negative,

a^{n}

is also non-negative. This means that the

n^{th}

root can be rewritten using a rational exponent.

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

Since both

n

and

\frac{1}{n}

are rational numbers, the Power of a Power Property for Rational Exponents can be applied to simplify the obtained expression.

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

▼

Simplify

PowPow

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

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

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

a^{1}

ExponentOne

$a^{1} = a$

a

Therefore,

n a^{n}

is equal to

a

a

is non-negative.

$a < 0$

In case of a negative value of $a,$ there are also two cases two consider.

$n$ is even
$n$ is odd

Even $n$

Recall that a root with an even index

n_{e}

is defined only for non-negative numbers. Although

a

is negative,

a^{n_{e}}

is positive. Also, a power with a negative base and an even exponent can be rewritten as a power with a positive base.

a^{n_{e}} = (- a)^{n_{e}}

Now, since

- a

is positive, the Power of a Power Property for Rational Exponents can be applied again to simplify

n_{e} a^{n_{e}} .

n_{e} a^{n_{e}}

▼

Simplify

NegBaseToPosPow

$(- a)^{n_{e}} = a^{n_{e}}$

n_{e} (- a)^{n_{e}}

RootToPowD

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

((- a)^{n_{e}})^{\frac{1}{n _{e}}}

PowPow

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

(- a)^{n_{e} \cdot \frac{1}{n _{e}}}

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

(- a)^{1}

ExponentOne

$a^{1} = a$

- a

Odd $n$

A root with an odd index

n_{o}

is defined for all real numbers. By the definition of the

n^{th}

root, the expression

n_{o} a^{n_{o}}

is the number

y

that, when multiplied by itself

n_{o}

times, will result in

a^{n_{o}} .

n_{o} times y \cdot y \cdot \dots \cdot y = a^{n_{o}}

Because

n_{o}

is odd and

a

is negative,

a^{n_{o}}

is also negative. This means that the best candidate for

y

is simply

a .

n a^{n} = a

Summary

If $a$ is non-negative, $n a^{n}$ is always equal to $a .$ However, in case of negative $a,$ the value of $n a^{n}$ depends on the parity of $n .$

	$a \geq 0$	$a < 0$
Even $n$	$n a^{n} = a$	$n a^{n} = - a$
Odd $n$	$n a^{n} = a$	$n a^{n} = a$

To conclude, for odd values of $n,$ the expression $n a^{n}$ is equal to $a .$ On the other hand, if $n$ is even, $n a^{n}$ can be written as $∣ a ∣ .$

n a^{n} = {∣ a, ∣ a ∣, if n is odd if n is even

Extra

Simplifying

n a^{m}

Depending on the index of the root and the power in the radicand, simplifying $n a^{m}$ may be problematic. Because real $n^{th}$ roots with an even index are defined only for non-negative numbers, the absolute value is sometimes needed.

If $n$ is even, $n a^{m}$ is defined only for non-negative $a^{m} .$

The following applet presents a decision tree to simplify

n a^{m} .

In this applet,

k

is the greatest common factor of

m

and

n .

Consider example

n^{th}

roots

n a^{m}

that can be simplified by using the decision tree.

$a$	$m$	$n$	$n a^{m}$	Simplify
$2$	$8$	$4$	$4 2^{8}$	$2^{\frac{8}{4}} = 2^{2} = 4$
$- 3$	$6$	$2$	$(- 3)^{6}$	$∣ ∣ ∣ ∣ (- 3)^{\frac{6}{2}} ∣ ∣ ∣ ∣ = ∣ ∣ ∣ (- 3)^{3} ∣ ∣ ∣ = ∣ - 27 ∣ = 27$
$x$	$6$	$3$	$3 x^{6}$	$x^{\frac{6}{3}} = x^{2}$
$- 3$	$2$	$8$	$8 (- 3)^{2}$	$4 ∣ - 3 ∣ = 43$
$2$	$3$	$6$	$6 2^{3}$	$2$
$- 2$	$3$	$9$	$9 (- 2)^{3}$	$3 - 2$
$2$	$6$	$8$	$8 2^{6}$	$4 2^{3} = 48$
$- 2$	$6$	$8$	$8 (- 2)^{6}$	$4 ∣ - 2 ∣^{3} = 4 2^{3} = 48$

Example

Homework on Radical Expressions

In addition to physics and astronomy, Ignacio is also interested in algebra. Unfortunately, he missed one class and he cannot solve his homework on radical expressions. Help Ignacio pair each expression to its simplified form.

Hint

Write the radicand as a power. Then, use the $n^{th}$ Roots of $n^{th}$ Powers Property to simplify the radical expression.

Solution

To simplify the given expressions, the

n^{th}

Roots of

n^{th}

Powers Property will be used.

n a^{n} = {∣ a, ∣ a ∣, if n is odd if n is even

Unfortunately, only the second expression is written in the form that allows to use the property right away. Because the power and the index of the root are both

5

and

5

is an odd number, the second expression simplifies to

x - 2 .

Radical Expression	Simplified Form
$x^{2} - 4 x + 4$	$?$
$5 (x - 2)^{5}$	$x - 2$
$4 16 x^{8}$	$?$
$4 x^{2}$	$?$

Each of the remaining expressions should be rewritten as a power with the exponent equal to the index of the corresponding root. Notice that the radicand in the first expression is a perfect square trinomial.

x^{2} - 4 x + 4 = x^{2} - 2 (2) x + 2^{2}

Now, the radical expression can be simplified by writing the radicand as the square of a binomial. Because the index of the root is an even number, the result will require the absolute value.

x^{2} - 2 (2) x + 2^{2}

FacNegPerfectSquare

$a^{2} - 2 a b + b^{2} = (a - b)^{2}$

(x - 2)^{2}

SqrtToAbs

$a^{2} = ∣ a ∣$

∣ x - 2 ∣

Therefore, the first expression simplifies to

∣ x - 2 ∣ .

Radical Expression	Simplified Form
$x^{2} - 4 x + 4$	$∣ x - 2 ∣$
$5 (x - 2)^{5}$	$x - 2$
$4 16 x^{8}$	$?$
$4 x^{2}$	$?$

Next, properties of exponents will be used to write the radicand

16 x^{8}

as a power with the exponent of

4 .

4 16 x^{8}

▼

Rewrite

WritePow

Write as a power

4 2^{4} x^{8}

SplitIntoFactors

Split into factors

4 2^{4} x^{2 \cdot 4}

ProdInExponent

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

4 2^{4} (x^{2})^{4}

ProdPowII

$a^{m} b^{m} = (a b)^{m}$

4 (2 x^{2})^{4}

$4 a^{4} = ∣ a ∣$

∣ ∣ ∣ 2 x^{2} ∣ ∣ ∣

Since

x^{2}

is non-negative,

2 x^{2}

is also non-negative. This means that

∣ ∣ ∣ 2 x^{2} ∣ ∣ ∣

is equal to

2 x^{2} .

Radical Expression	Simplified Form
$x^{2} - 4 x + 4$	$∣ x - 2 ∣$
$5 (x - 2)^{5}$	$x - 2$
$4 16 x^{8}$	$2 x^{2}$
$4 x^{2}$	$?$

Now the final expression

4 x^{2}

will be examined. Similar to the previous expression, properties of exponents will be used to rewrite the radicand as a square.

4 x^{2}

▼

Simplify

WritePow

Write as a power

2^{2} x^{2}

ProdPowII

$a^{m} b^{m} = (a b)^{m}$

(2 x)^{2}

SqrtToAbs

$a^{2} = ∣ a ∣$

∣ 2 x ∣

This time, it is not known whether

2 x

is non-negative. Therefore, the absolute value is needed in this expression. Finally, all simplified forms may be written in the table.

Radical Expression	Simplified Form
$x^{2} - 4 x + 4$	$∣ x - 2 ∣$
$5 (x - 2)^{5}$	$x - 2$
$4 16 x^{8}$	$2 x^{2}$
$4 x^{2}$	$∣ 2 x ∣$

Discussion

Product and Quotient Properties of Radicals

Usually, the $n^{th}$ Roots of $n^{th}$ Powers Property is not enough to simplify radical expressions. Therefore, more properties will be presented and proven in this lesson. The first one refers to the root of a product.

Rule

Product Property of Radicals

Given two non-negative numbers $a$ and $b,$ the $n^{th}$ root of their product equals the product of the $n^{th}$ root of each number.

$n a b = n a \cdot n b,$ for $a \geq 0$ and $b \geq 0$

If $n$ is an odd number, the $n^{th}$ root of a negative number is defined. In this case, the Product Property of Radicals for negative $a$ and $b$ is also true.

Proof

First, a special case of the property will be proven. Assume that

a

b

is equal to

0 .

By the Zero Property of Multiplication, the radicand in left-hand side of the formula is equal to

0 .

\underline{Left - Hand Side} n 0 = ? \underline{Right - Hand Side} n a \cdot n b

Because

0^{n} = 0,

n^{th}

root of

0

is equal to

0 .

This means that both the left-hand side and one of the factors on the right-hand side equal

0 .

Again, by the Zero Property of Multiplication, the right-hand side is also

0 .

\underline{Left - Hand Side} 0 = \underline{Right - Hand Side} 0

Therefore, the property is true when

a

b

is equal to

0 .

Next, assume that both

a

and

b

are nonzero. Let

x,

y,

and

z

be real numbers such that

x = n a,

y = n b,

and

z = n a b .

By the definition of an

n^{th}

root, each of these numbers raised to the

n^{th}

power is equal to its corresponding radicand.

⎩ ⎪ ⎪ ⎨ ⎪ ⎪ ⎧ x^{n} = a y^{n} = b z^{n} = a b (I) (II) (III)

Next, because

b

is not equal to

0,

Equation (I) can be multiplied by

y^{n},

which is equal to

b .

x^{n} = a ⇓ x^{n} y^{n} = a y^{n}

Now, substitute Equations (II) and (III) into the obtained equation.

x^{n} y^{n} = a y^{n}

▼

Substitute values and simplify

Substitute

$y^{n} = b$

x^{n} y^{n} = a b

Substitute

$a b = z^{n}$

x^{n} y^{n} = z^{n}

ProdPowII

$a^{m} b^{m} = (a b)^{m}$

(x y)^{n} = z^{n}

RearrangeEqn

Rearrange equation

z^{n} = (x y)^{n}

Since

x y

and

z

are of the same sign, the final equation implies that

z = x y .

z^{n} = (x y)^{n} \Rightarrow z = x y

The last step is substituting

z = n

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Catch-Up and Review

Proof

a≥0

a<0

Even n

Odd n

Summary

Extra

Hint

Solution

Proof

$a \geq 0$

$a < 0$

Even $n$

Odd $n$