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Algebra 1

5.2.2 The nth Root

Algebra 15.2.2 The nth Root

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Activity

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

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

Expression Process Simplified Result
n n ( 2 ) ( 2 ) -2
n 2 n 2 ( 2 ) 2 = ( 2 ) ( 2 ) ( 2 ) 2 = ( 2 ) ( 2 ) 4
n 3 n 3    
n 4 n 4    
n 5 n 5    

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

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

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

The nth root, or radical, is written a n a n . If b n = a b n = a , then b b is an nth root of a a . The n n is called the index of the radical. Just like n 2 n 2 is called the square root, n 3 n 3 is the cubed root, n 4 n 4 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 x y = z z y = x z y = x
4 3 = 64 4 3 = 64 64 3 = 4 64 3 = 4
3 4 = 81 3 4 = 81  
  32 5 = 2 32 5 = 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 27 3 .
  1. -9
  2. -3
  3. 9
  4. 3

Additional Resources

Simplifying Radicals

A labeled diagram showing the parts of a radical: the index (3), the radical symbol, and the radicand (8). The term radical is labeled to encompass the entire expression.

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 × 2 × 2 = 8 2 × 2 × 2 = 8 , so 8 3 = 2 8 3 = 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

Try It: Simplifying Radicals

Simplify.

32 5 32 5

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