Exponents and Roots


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You may not realize it yet, but exponents and roots are two of the easiest things to learn in math. There's not a whole lot to it. Exponents and roots are the opposites of each other. If 4 raised to the power of 2 equals 16, then the square root of 16 is 4. A square root has a root index of 2.

Looking at the picture above, notice the terms in yellow for exponents. The exponent or power is a number that tells us how many times to multiply the base by itself. In the picture above we have 3 to the third power. This simply means to multiply the base, which is 3, by itself 3 times. In other words, 3x3x3=27. If the exponent had been 4 instead of 3, we would have had to multiply the base, which is 3, by itself 4 times like this 3x3x3x3=81. Pretty easy huh?

Let's take another look at the picture above, this time looking at the terms in yellow for roots. The root index tells us which root to find for the number under the radical sign. If the root index is 2, then it is usually not shown. Roots with indexes of 2 are called "Square roots." You'll probably work with square roots more than anything else. In the picture above we have the number 27 under the radical sign. Our root index is 3. Roots with indexes of three are normally called "cubed roots." This tells us that we need to figure out which number multiplied by itself 3 times gives us 27. As we already saw earlier, 3 raised to the power of 3 equals 27, so the cubed root of 27 equals 3. Remember that cubed roots have a root index of 3. Square roots have a root index of 2, which is usually not shown. What if the number 81 had been under the radical sign and the root index had been 4? Think back to what we did earlier. Since the root index is 4 this time, we want to find a number that when multiplied by itself 4 times will equal 81. If you remember 3x3x3x3=81 so the fourth root of 81 equals 3.


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The picture above gives several examples of what happens when you raise 2 to different powers. 2 raised to the 0 power equals 1. This is true for any number except 0. 0 CAN NOT be raised to the 0 power. If you were to raise 2,000 to the 0 power, you'd still get 1 for the answer. Negative numbers raised to the 0 power also equal 1. 2 raised to the power of 1 is 2. Any number, even 0, raised to the power of 1 will equal that number. 2000 raised to the power of 1 is 2000. 0 raised to the power of 1 is 0. -45 raised to the power of 1 is -45. 2 raised to the power of 2, or "two squared," means 2x2. 2 raised to the power of 3 means 2x2x2. 2 raised to the power of 6 means 2x2x2x2x2x2. In other words, powers greater than 1 mean to multiply the base number by itself the amount of times equal to whatever the exponent is.

It's important to remember that when raising negative numbers to a power, the rules for multiplying negative numbers must be followed. For instance, -2 raised to the power of 2 is -2x-2=4. -2 raised to the power of 3 is -2x-2x-2=-8. When raising negative numbers to a power, if the exponent is an even number, the answer will always be positive. If the exponent is an odd number, the answer will always be negative.


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Above is a picture with several examples of roots that equal 2. Notice the first example doesn't have a root index. Since there is no root index, we can safely say that the root index is 2. As we mentioned earlier, when the root index is 2 we usually leave it off. We also mentioned that root indexes of 2 are known as a "square roots." The picture shows that the square root of 4 is 2. The third root of 8 is 2, because 2x2x2=8. We could have also said that the "cubed root" of 8 is 2. The sixth root of 64 is also 2, since 2x2x2x2x2x2=64.

When you're trying to find the root of a negative number, make sure the root index is an odd number like 3,5,7, and so forth. If you're looking for the root of a negative number and the root index is an even number, there is no root to be found! Remember that even numbers are 2,4,6,8, and so on. It is IMPOSSIBLE to find a root for a negative number with a root index that is even. Why? Think about negative numbers for a minute. -1x-1=1. -1x-1x-1=-1. -1x-1x-1x-1=1. -1x-1x-1x-1x-1=-1. ONLY when you multiply a negative number by itself an odd amount of times will the answer ever be negative. There is NOT anything that you can multiply by itself an even amount of times that will give a negative number for the answer. The answer will always be positive.

Below is a picture comparing a few exponent and root opposites. Please look it over carefully. When you're ready, you can take the quiz to see how much you've learned. Thank you very much for sitting through this lesson! Hopefully you've found that exponents and roots are two of the easier things to learn in the mathematics world.


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QUIZ

Question 1:

5 raised to the second power = _____

Answer:
A:  2
B:  5
C:  25
D:  50

Question 2:

2 raised to the third power = _____

Answer:
A:  3
B:  4
C:  6
D:  8

Question 3:

3 raised to the fourth power = _____

Answer:
A:  12
B:  27
C:  64
D:  81

Question 4:

0 raised to the first power = _____

Answer:
A:  0
B:  1
C:  -1
D:  The problem has no answer

Question 5:

16 raised to the zero power = _____

Answer:
A:  0
B:  1
C:  16
D:  The problem has no answer

Question 6:

The square root of 16 = _____

Answer:
A:  2
B:  4
C:  16
D:  256

Question 7:

The cubed root of 27 = _____

Answer:
A:  3
B:  9
C:  27
D:  19683

Question 8:

The fourth root of 16 = _____

Answer:
A:  2
B:  4
C:  8
D:  None of the above

Question 9:

The fifth root of 1 = _____

Answer:
A:  0
B:  1
C:  5
D:  None of the above

Question 10:

The sixth root of -64 = _____

Answer:
A:  0
B:  -2
C:  2
D:  The problem has no answer




~ Other Links ~

Exponents - First Glance

Square Roots - First Glance

SparkNotes - Exponents

Ask Dr. Math - Exponents

Ask Dr. Math - Square Roots

Harcourt E-Lab

Harcourt Math Glossary


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