Lynn Marecek; MaryAnne Anthony-Smith; Andrea Honeycutt Mathis; Jay Abramson; Sharon North; Amy Baldwin; Alyssa Howell

5.2.2 The nth Root

5.2.2 • The nth Root

Activity

Work with a partner to discuss and answer the following questions.

1. Complete the table when $n = - 2$ on your paper and check your answers.

Expression	Process	Simplified Result
$n$	$(- 2)$	-2
$n^{2}$	$(- 2)^{2} = (- 2) (- 2)$	4
$n^{3}$
$n^{4}$
$n^{5}$

Compare your answers:

Expression	Process	Simplified Result
$n$	$(- 2)$	-2
$n^{2}$	$(- 2)^{2} = (- 2) (- 2)$	4
$n^{3}$	$(- 2)^{3} = (- 2) (- 2) (- 2)$	-8
$n^{4}$	$(- 2)^{4}$	16
$n^{5}$	$(- 2)^{5}$	-32

2. What do you notice about the answers when the powers are even?

Compare your answer:

They are positive.

3. What do you notice about the answers when the powers are odd?

Compare your answer:

They are negative.

The $\sqrt{a}$ , or square root of $a$ , is the same as $\sqrt[2]{a}$ . What about when we want to take roots that are higher than the square root? These are called radicals.

The $n^{th}$ root, or radical, is written $\sqrt[n]{a}$ . If $b^{n} = a$ , then $b$ is an $n^{th}$ root of $a$ . The $n$ is called the index of the radical. Just like $\sqrt[2]{n}$ is called the square root, $\sqrt[3]{n}$ is the cubed root, $\sqrt[4]{n}$ is the fourth root, and so on. The number inside the radical is called the radicand.

4. Using what you did with the powers in number 1, complete the table to find the roots of each.

$x^{y} = z$	$\sqrt[y]{z} = x$
$4^{3} = 64$	$\sqrt[3]{64} = 4$
$3^{4} = 81$
	$\sqrt[5]{- 32} = - 2$

Compare your answers:

$x^{y} = z$	$\sqrt[y]{z} = x$
$4^{3} = 64$	$\sqrt[3]{64} = 4$
$3^{4} = 81$	$\sqrt[4]{81} = 3$
${(- 2)}^{5} = - 32$	$\sqrt[5]{- 32} = - 2$

Could we have an even root of a negative number? We know that the square root of a negative number is not a real number. The same is true for any even root. Even roots of negative numbers are not real numbers. Odd roots of negative numbers are real numbers.

Self Check

Simplify

\sqrt[3]{-27}

.

-9
-3
9
3

Additional Resources

Simplifying Radicals

When simplifying radicals, think which number will be used as a factor the number of times indicated by the index to get the number under the radical.

In the example above,

$2 \times 2 \times 2 = 8$ , so $\sqrt[3]{8} = 2$ .

Remember, when the index is odd, the root is allowed to be negative. When the index is even, the root cannot be negative.

Try it

Simplifying Radicals

Simplify.

$\sqrt[5]{- 32}$

Compare your answer:

Here is how to simplify a radical:

Step 1 - Identify the index.

5

Step 2 - Identify the radicand (inside the radical).

–32

Step 3 - What number can be used as a factor 5 times to get –32?

$(- 2) \times (- 2) \times (- 2) \times (- 2) \times (- 2) = - 32$

Step 4 - Simplify.

$\sqrt[5]{- 32} = - 2$