Exploring Operations with Radical Expressions

	$n$ 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, $n a$

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

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

a

n

n a^{n}

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

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

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

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

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

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

Radical Expression

Simplified Form

x^{2} - 4 x + 4

?

5 (x - 2)^{5}

x - 2

4 16 x^{8}

?

4 x^{2}

?

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

Radical Expression

Simplified Form

x^{2} - 4 x + 4

∣ x - 2 ∣

5 (x - 2)^{5}

x - 2

4 16 x^{8}

?

4 x^{2}

?

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

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}

?

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

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 ∣

Simplify the given radical expressions.

81 x^{4} y^{8}

8 256 a^{24}

According to n^\text{th} Roots of n^\text{th} Powers Property, for any real number a, the radical expression sqrt(a^n) can be simplified as follows. sqrt(a^n)= a, & if n is odd |a|, & if n is even Because the given root has an even index and the variables have even exponents, we will use the absolute value symbol to simplify the expression.

Any number raised to an even exponent will always be non-negative. The expression 9x^2y^4 will always be non-negative because there are only even exponents. The product of non-negative numbers is also always non-negative. Therefore, the absolute value symbol is not needed in this case. |9x^2y^4| ⇔ 9x^2y^4

Just as in Part A, n^\text{th} Roots of n^\text{th} Powers Property will be applied. sqrt(a^n)= a, & if n is odd |a|, & if n is even Because the given root has an even index and the variable has an even exponent, we will use the absolute value symbol to simplify the expression.

Since the exponent of a is odd, it is necessary to use the absolute value symbol because the expression 2a^3 could be negative for negative values of a.

Use the Properties of Radicals to simplify the given radical expression.

4 3 π^{2} \cdot 4 9 π^{2} \cdot 43

3 125 a^{6} b^{3} c^{9}

To simplify the given expression, we will use the Properties of Radicals. Let's start by applying the Product Property of Radicals.

Next, use the Product of Powers Property to multiply the π-factors.

To simplify the given expression, we will use the Properties of Radicals. We will start by factoring out perfect cubes and using the Product Property of Radicals. Moreover, consider the following two cases for any real number a. sqrt(a^n)= a, &if n is odd |a|, &if n is even Because our radical has an odd index, we will not need to use absolute value symbols to simplify our expression.

Simplify the given rational expression. Make assumptions on the variables

x,

y,

and

z

so that the square roots are real.

\frac{7 5 x ^{3} y z ^{3}}{4 8 x \cdot z ^{5}}

Let's start by rewriting the given expression by applying the Product Property of Radicals. sqrt(75x^3yz^3)/sqrt(48x)* sqrt(z^5) = sqrt(75x^3yz^3)/sqrt(48xz^5) Next, before we can divide this expression, we need to answer three questions.

What are the assumptions on the variables?
Can the expressions be divided?
Do absolute value symbols need to be added to the answer?

Let's start with the first question. For the square roots to be real, their radicands should be non-negative because otherwise, the square roots would be imaginary. The radicand in the denominator is non-negative only if both x and z are non-negative or both x and z are non-positive. Denominator [0.1cm] 48xz^5≥ 0 ⇔ ( x, z≥ 0 or x, z ≤ 0 ) Recall that the square roots in the denominator had been separated before applying the Product Property of Radicals. This means that both x and z must be non-negative. For the square root in the numerator to be also real, the radicand 75x^3yz^3 should be non-negative. Since x and z are non-negative, y must be non-negative as well. Numerator [0.1cm] 75x^3yz^3≥ 0 x, z≥ 0 ⇒ y≥ 0 Therefore, all three variables must be non-negative. Now let's answer the second question. The expressions can be divided only if the denominator is not equal to 0. This puts an additional restriction on the x- and z-variables, as neither can be equal to 0. Since we already established that these variables are non-negative, this implies that x and z must be positive. x, z≥ 0 x, z≠ 0 ⇒ x, z>0 Finally, to answer the third question, consider the rule regarding absolute value symbols for any real number a. sqrt(a^n)= a, & if n is odd |a|, &if n is even In our case, the index is 2, an even number. According to the rule, we need to use absolute value bars in our answer. However, we also found that all of the variables can only take non-negative values. Since the absolute value of a non-negative value is the value itself, we do not need absolute value symbols after all. Let's sum up all the information we have found.

Question	Answer
What are the assumptions on the variables?	x, y, z≥ 0
Can the expressions be divided?	Yes, if x, z≠ 0.
Do absolute value symbols need to be added to the answer?	No, as x, z>0 and y≥ 0.

Now we can use the Quotient Property of Radicals and reduce the fraction.

Next, let's simplify the radical expression by finding all of the perfect squares inside the radicals. Then we will apply the n^\text{th} Roots of n^\text{th} Powers Property.

Rationalize the denominator of the following radical expressions and simplify if necessary.

\frac{2 2 - 4 6}{2 + 2 6}

4 \frac{9 x ^{3} y ^{2}}{4 a ^{3} b}

We need to rationalize a fraction with an irrational binomial denominator. To do so, we multiply the numerator and denominator by the irrational conjugate of the denominator. We find the conjugate by changing the sign of the second term of the radical expression.

Binomial	Conjugate
asqrt(c) + bsqrt(d)	asqrt(c) - bsqrt(d)
asqrt(c) - bsqrt(d)	asqrt(c) + bsqrt(d)

In this case, the conjugate of the denominator is sqrt(2)-2sqrt(6).

Now that the denominator has been rationalized, the expression can be further simplified.

To simplify the given expression, we can first rewrite the radical as a quotient of two radicals. sqrt(9x^3y^2/4a^3b) = sqrt(9x^3y^2)/sqrt(4a^3b) Now, we can rationalize the denominator of the quotient. To do so, we will first multiply the numerator and denominator by the irrational conjugate of an n^\text{th} root and use the fact that we can multiply the radicands of n^\text{th} roots if they have the same index.

If sqrt(a) and sqrt(b) are real numbers, then sqrt(a)* sqrt(b) is equal to sqrt(ab).

Let's start by finding the exponents necessary to create perfect 4^(th) powers in the denominator. Our goal is to have each factor raised to the 4^\text{th} power.

Now that we have found the factors that make the radicand of the denominator perfect 4^(th) powers, we can begin to simplify the quotient. Consider the index of the radicals to see how we should format our solution. The n^\text{th} Roots of n^\text{th} Powers Property can help with this. sqrt(a^n)= a, &if n is odd |a|, &if n is even Since both radicals are real numbers and the roots are even, the expressions underneath the radicals must be positive & mdash; if they were not, the radicals would be imaginary. With this in mind, let's consider the possible values of the variables a, b, x, and y.

Before expanding the fraction, the numerator consisted of x to an odd power. This means that x must be non-negative.
The index in the numerator is even and the exponents of a and b are odd. Therefore, in order for this radical expression to result in a real number, a and b must be of the same sign.
The index in the numerator is even and the exponent of y is even. Therefore, the expression will be real whether the value of y is positive or negative.
Since both a and b are in the denominator, they must be nonzero.

This means that to calculate the fourth root of a or b, we will need absolute value symbols.

Simplify the following radical expression.

3332 + 3500 - 33108 + 23128

To simplify the given radical expression, we first need to rewrite the radicands so that they have exponents that match the index of the radicals. Then we will consider the properties for combining radical expressions when they are part of a sum or difference. asqrt(x)+bsqrt(x)=(a+b)sqrt(x) asqrt(x)-bsqrt(x)=(a-b)sqrt(x) Notice that radicals can only be added or subtracted when the index and the radicand are exactly the same. Let's simplify our radicals to see if we can create like terms.

Now three terms have the same radical expression. Let's add and subtract them!

Now, let's consider the remaining radicals. 2sqrt(4) + 8sqrt(2) Because the radicands are not the same, it is not possible to add them. Therefore, the simplest form of the radical expression is 2sqrt(4) + 8sqrt(2).

	15 Theory slides
	11 Exercises - Grade E - A
	Each lesson is meant to take 1-2 classroom sessions

Operations With Radical Expressions

Catch-Up and Review

Proof

a≥0

a<0

Even n

Odd n

Summary

Extra

Hint

Solution

Proof

Proof

Hint

Solution

Hint

Solution

Hint

Solution

Hint

Solution

Proof

Hint

Solution

Operations With Radical Expressions

Recommended exercises

$a \geq 0$

$a < 0$

Even $n$

Odd $n$