Building Simple Words: Exponents and Roots

Multiplication is multiple additions. Exponentiation is multiple multiplications. Exponents are written as superscript numbers above and to the right of the number to be exponentiated. 2×2×2 is written as 2³ and equals 8. This is read as 2 raised to the 3rd power. Not all exponents can be shown in this format, so I will use a carat (^) to indicate the number that follows is the exponent.

Numbers raised to the second power are said to be squared because the area of a square is the product of the two equal dimensions (length, width) multiplied together. Numbers raised to the third power are said to be cubed because the volume of a cube is the product of the three equal dimensions (length, width, height) multiplied together. All other exponents use the ordinal form of the exponent (to the fourth, to the fifth, to the sixth, ect.) Any number raised to the 0 power equals 1.

Squares are so common that the squares from 1 to 10 should be memorized. They are taken from the multiplication table.

1² = 1×1 = 1
2² = 2×2 = 4
3² = 3×3 = 9
4² = 4×4 = 16
5² = 5×5 = 25
6² = 6×6 = 36
7² = 7×7 = 49
8² = 8×8 = 64
9² = 9×9 = 81
10² = 10×10 = 100

You can't directly add numbers with exponents. Each must be exponentiated first, then the results are added together. With multiplication, if the numbers are the same the exponents are added together. 2^3×2^4=2^(3+4)=2^7. If you exponentiate an exponent, you multiply the exponents. (2^3)^4=2^(3×4)=2^12. With a rational number, when the number in the numerator and the number in the denominator are the same, you subtract the exponents. 2^4/2^3=2^(4-3)=2^1=2. If the exponent in the numerator is larger, you subtract the exponent in the denominator from the exponent in the numerator and replace the denominator with 1. 2^6/2^3=2^(6-3)/1=2^3/1=2^3. If the exponent in the denominator is larger, you subtract the exponent in the numerator from the exponent in the denominator and replace the numerator with 1. 2^3/2^6=1/2^(6-3)=1/2^3.

Just as division is the opposite of multiplication, there is an opposite operation for exponentiation. The root of a number is the number that when multiplied by itself gives the orignal number. Roots are indicated with a symbol called a surd √ with an index over the little bar. The index shows the root and the surd separates the number to be broken down. The index and surd together with a number are called a radical. If the root is the square root the index is not shown. ³√27 = 3 because 3³=27.

Like with addition of exponents, you can't add different radicals. You have to get the roots first, then add the resulting numbers. We can mix numbers and radicals. A number can be added to a radical or a radical can be multiplied by a number. 3+√5 is a valid number as is 3√5. We can add two radicals together if they are exactly the same. In that case, you add the numbers on the outside and keep the radical unchanged. 4√5 + 3√5 = 7√5. I can't add 3√25 and 4√36 because the radicals are different. I would have to convert the radicals to numbers then add the result. 3√25+4√36=3(5)+4(6)=15+24=39.

Roots can be expressed as rational exponents. A square root is a ½ exponent. √4=4^½. If the number inside the radical is raised to a power, the power inside the radical becomes the numerator of the rational exponent and the index of the radical becomes the denominator of the rational exponent. Converting roots to exponents means we can use our rules for multiplying and dividing exponents to multiply two roots together. If the numbers inside the radical are the same we simply add the exponents together using the rules for adding and subtracting rational numbers. If the exponent on the number under to surd is the same as the index of the root, the index and the exponent cancel each other out and the surd disappears. ³√27³=27^(3/3)=27^1=27. If the indexes are the same, multiply the numbers under the surd and keep the index. ³√9׳√8=³√(9×8)=³√72.

To simplify a radical you remove groups of common factors based on the index. First you factor the number under the surd into its prime factors. Then you remove groups of factors based on the index. When all of the groups have been removed, you multiply the numbers on the outside together and multiply the numbers remaining under the radical. Let's simplify ³√72. The prime factors of 72 are 3×3×2×2×2. The index is 3 so we want to remove factos in groups of 3. There are 3 2's so we remove them from inside the surd and place a single 2 on the outside;2³√(3×3). There are only 2 3's under the surd and we need to pull out in groups of three, so the 3's stay under the surd. Multiply the 3's back together again and ³√72=2 ³√9. How about ³√1296. First we factor 1296 into its prime factors; ³√(3×3×3×3×2×2×2×2). We need to remove factors in groups of 3. There are 4 3's, so we can remove 3 of them and there are 4 2's so we can remove 3 of them as well; (3)(2)³√(3×2). There are no more groups of 3 left so we multiply the numbers on the outside and multiply the numbers on the inside; 6 ³√6.

Roots of prime numbers don't have an even representation because there are no factors of prime numbers, so you can't get a prime number by multiplying any number to itself. All roots of prime numbers are irrational numbers. We'll discuss them next.

Even roots of negative numbers are also a problem. If you think about the rules for multiplying negative numbers, you can see where the problem lies. Remember that if there are an even number of negative numbers being multiplied together, you get a positive number. If there are an odd number of negative numbers, you get a negative number. A negative number exponentiated an even number of times gives a positive number. So there is no way to get a negative number if the exponent is even, thus there can't be an even root of a negative number. Finding an even root of a negative number requires us to use a very special number called the imaginary number. We won't worry about the imaginary number for now.

Modern calculators usually have a key for finding any root of any number, but for a method for finding any root of any number see this article in Wikipedia Nth Root Algorithm. Now that we know about roots, let's talk about the last group of numbers that make up our alphabet.