Problem 1 Finding All Real Roots

1. What are the real cube roots of 0 . 008 , − 1000 , 0 . 008 comma negative , 1000 , comma  and 1 27 1 twenty seventh ?

   0.008 = ( 0.2 ) 3 0.2  is the only real cube root of  0.008. − 1000 = ( − 10 ) 3 − 10  is the only real cube root of  − 1000 . 1 27 = ( 1 3 ) 3 1 3  is the other real fourth root of  1 27 . table with 3 rows and 3 columns , row1 column 1 , 0.008 , column 2 equals . open 0.2 close cubed , column 3 0.2 . is the only real cube root of . 0.008. , row2 column 1 , negative , 1000 , column 2 equals . open negative 10 close cubed , column 3 negative 10 . is the only real cube root of . minus , 1000 , . , row3 column 1 , 1 twenty seventh , column 2 equals . open , 1 third , close cubed , column 3 1 third . is the other real fourth root of . 1 twenty seventh , . , end table

   Plan

   How many real cube roots are there?

   A cube root is the same as a third root, and 3 is odd. So there is only one real cube root of a number.

2. What are the real fourth roots of 1, − 0.0001 , negative , 0.0001 , comma  and 16 81 ? 16 over 81 , question mark

   Since 1 is positive, there are two real fourth roots.

   1 = 1 4 1  is a real fourth root of  1 . 1 = ( − 1 ) 4 − 1  is the other real fourth root of  1. table with 2 rows and 2 columns , row1 column 1 , 1 equals , 1 to the fourth , column 2 1 . is a real fourth root of . 1 . , row2 column 1 , 1 equals . open negative 1 close to the fourth , column 2 negative 1 . is the other real fourth root of . 1. , end table

Since − 0.0001 negative , 0.0001  is negative, there are no real fourth roots of − 0.0001 . negative , 0.0001 , .

Since 16 81 16 over 81  is positive, there are two real fourth roots.

16 81 = ( 2 3 ) 4 2 3 is a real fourth root of 16 81 . 16 81 = ( − 2 3 ) 4 − 2 3 is the other real fourth root of 16 81 . table with 2 rows and 2 columns , row1 column 1 , 16 over 81 , equals . open , 2 thirds , close to the fourth , column 2 2 thirds . isarealfourthrootof . 16 over 81 , . , row2 column 1 , 16 over 81 , equals . open , negative , 2 thirds , close to the fourth , column 2 negative , 2 thirds . istheotherrealfourthrootof . 16 over 81 , . , end table

✓ Got It?

1. 1. What are the real fifth roots of 0, − 1 , negative 1 comma  and 32?
   2. What are the real square roots of 0.01, − 1 , negative 1 comma  and 36 121 ? 36 over 121 , question mark
   3. Reasoning Explain why a negative real number b has no real nth roots if n is even.

Problem 2 Finding Roots

What is each real-number root?

1. − 8 3 cube root of negative 8 end root ,

   Plan

   How can you find a cube root?

   Work backwards. Find a number whose cube is the radicand.

   ( − 2 ) 3 = − 8 open negative 2 close cubed . equals negative 8

   So, − 8 3 = − 2 . cube root of negative 8 end root , , equals negative 2 .

2. 0.04 square root of 0.04

   ( 0.2 ) 2 = 0.04 open 0.2 close squared . equals , 0.04

   So, 0.04 = ◄ ( − 0.2 ) 2 = 0.04  also, but 0.04  represents the positive square root . square root of 0.04 , equals black left pointing pointer . begin box , table with 3 rows and 1 column , row1 column 1 , open , negative 0.2 , close squared . equals , 0.04 . alsocommabut , row2 column 1 , square root of 0.04 . representsthe , row3 column 1 , positivesquareroot . . , end table , end box

3. − 1 4 the fourth , root of negative 1 end root ,

   There is no real root because there is no real number whose fourth power is − 1 . negative 1 .

4. ( − 2 ) 2 square root of open , negative 2 , close squared end root

   ( − 2 ) 2 = 4 = 2 . square root of open , negative 2 , close squared end root . equals square root of 4 equals 2 .