Exponents and Radicals

An exponent represents repeated multiplication. Just as 2*2*2 can be written as 2³, x*x*x*x can be written as x⁴. The exponent is the 4. The base is the x. Also, x⁴ is called "x to the power of 4."

A negative exponent means the number is dividing into something:

	x-1	=	1 x
	x-2	=	1 x2

Any number raised to the power of 0 becomes one:

2⁰ = 1
x⁰ = 1

Any number raised to the power of one produces the same number:

9¹ = 9
x¹ = x

One of the first rules of exponents is the multiplication rule. If two numbers with the same base are multiplied together, their exponents will add. If x⁴ is multiplied with x², in the form of x⁴x², the result will be the base with the exponents added which would be x⁶. For example:

x⁴x² = x⁶
(x*x*x*x)(x*x) = x*x*x*x*x*x
x*x*x*x*x*x = x*x*x*x*x*x

Another rule of exponents is the division rule. When x⁶ is divided by x³. The exponents subtract and the remaining base stays the same.

x*x*x*x*x*x x*x*x

=

x*x*x

x*x*x  =  x*x*x

or since 1/x³ is the same as x^-3, we can write the problem as:

x⁶x^-3  =  x^(6-3)  =  x³

Any number with exponents contained inside of parentheses have their exponents multiplied by the exponent outside of the parentheses.

(x⁴)²  =  x⁴*x⁴  =  x⁸
(x⁴)²  =  x^(4*2)  =  x⁸

Sometimes the base can have more than one term. In this case, it is contained within parentheses with the exponent outside.

(x+1)²

(x+1)²  =  (x+1)(x+1)

As long as the base is the same the exponents can combine.

Using the division rule for exponents this would be:

The root of a basic number is another number which if multiplied by itself a certain number of times will result in that basic number. For example, if:

a*a  =  x  or  a*a*a  =  x  or  a*a*a*a  =  x

then a is a root of x.

One of the ways a root of a number can be symbolized is by ⁿ√x. Where √ is the radical symbol.

Also,

³√x * ³√x * ³√x  =  (³√x )³  =  x.

Taking the square root of the square of a number will result in that number:

²√x²  =  x

or

√x²  =  x

Taking the second root of a number is given by just the √ with no root number.

As demonstrated above, roots can be used to cancel out exponents. The same rules work with expressions inside the radical sign.

√(x+1)²  =  (x+1)
√(x+1)⁴ = √((x+1)²)²  =  (x+1)²

The radical cancels out the outer exponent of the expression.

Radicals can also be written as exponents. For example:

^a√x^b  =  x^b/a

examples

³√x  =  x^1/3
5√x⁷  =  x^7/5

The exponent becomes the numerator and the root becomes the denominator. The same applies to expressions:

⁴√(x+1)³  =  (x+1)^3/4