Terminal Information

What are Exponents?

Summary:

xⁿ = x * x * ...x (n times)

Exponents represent how many times a term (a variable or number) is multiplied by itself. For example:

x³
Is the same as:
x * x * x

Adding Variables With Exponents

Summary:

x^a + x^a + x^b + x^b = 2x^a + 2x^b
(Note: You can not add terms with different exponents.)

Just like with adding any other like terms you can "condense" terms with exponents, or represent adding the same thing multiple times though multiplication. For example:

x + x
Is the same as:
2x
and
x^a + x^a
Is the same as:
2x^a

We can see this is true if we were to substitute x for five:

5 + 5 = 10
Is the same as:
2(5) = 10
Just like:
5² + 5² = 25 + 25 = 50
is the same as:
5² + 5² = 2(5²) = 2(25) = 50

In order to combine terms the exponents of the terms must be the same. If you have multiple terms, some with the same exponents, and some not, then you can add the ones with the same exponents, but not the others. For example:

x² + 3x² + 2x² + 2x³ + x³ + 4x³ + x⁵ + 7x⁶
Is the same as:
6x² + 7x³ + x⁵ + 7x⁶

In the example above, 6x² + 7x³ + x⁵ + 7x⁶ is as reduced as the equation can be because we cannot add the terms with different exponents. While the terms may be the same, the numbers represented as a whole are very different when the exponents do not match.

This difference becomes obvious when you substitute in any number (barring zero or one) for the term. For example five raised to the second power (5² = 25) is very different from five raised to the third power (5³ = 125). If you attempt to combine these you soon find that the rules for doing so would be ambiguous. Which exponent would you use for the combined term? These problems become evident when trying to add 5's with different exponents:

5² + 5³ = 5² + 5³ = 5² + 5³
↓
5² + 5³ ≠ 2(5²) ≠ 2(5³)
↓
25 + 125 ≠ 2(25) ≠ 2(125)
↓
150 ≠ 50 ≠ 250

Product Rule (Multiplication)

Summary:

x^a * x^b = x^{a + b}

When multiplying same terms together, you can simply add the exponents. Remember, exponents are a way of telling you how many times you are multiplying a term by its self. You can think of this as the count of a "group". If you then multiply one group by another group of the same term, you are essentially just making the count of that "group" larger.

x² * x³ = x⁵
Is the same as:
(x * x) * (x * x * x) = x * x * x * x * x

We can see this example works if we replace x with 3:

3² * 3³ = 3⁵
↓
9 * 27 = 243
↓
243 = 243
Is the same as:
(3 * 3) * (3 * 3 * 3) = 3 * 3 * 3 * 3 * 3
↓
(9) * (9 * 3) = 9 * 9 * 3
↓
(9) * (27) = 81 * 3
↓
243 = 243

Quotient Rule (Division)

Summary:

Similar to multiplication, division of like terms with exponents can be calculated by subtracting the divisor's exponent from the dividend's exponent. Remember that a number divided by itself is equal to one, and multiplying a number by one does nothing. Since when you are dividing two like terms with exponents, you are essentially dividing two "groups" of the same numbers, the terms in one group negate the effects or "cancel out" the terms in the other group.

Negative Exponents

Summary:

Negative exponents are actually the inverse "fraction" of the term to the absolute value of the exponent. That is in other words one divided by the term to the (non-negative) exponent.

We can see how this "inverse" is actually the same as a negative exponent when using the quotient rule.

Power Rule

Summary:

(x^a)^b = x^{a * b}

When one term raised to an exponent is raised to another exponent, you multiply the exponents together. Remember that and exponent is essentially just saying, "A thing times its self this number of times". In the case of the exponent being outside the parenthesizes, the thing multiplied by itself is everything in the parenthesizes! This becomes evident when you write it out.

(x²)³ = x^{2 * 3} = x⁶
Is the same as:
(x²)³ = (x²) * (x²) * (x²) =
(x * x) * (x * x) * (x * x) =
x * x * x * x * x * x

Factors and Roots

Factors are sort of the "DNA" of numbers. They are the numbers when multiplied together make up a number. Any integer number can be divided into its factors. A number's factors can also be one and itself. For example the number 15's factors are 1 and 15, as well as 3 and 5. If a number's factors are only one and itself, then the number is a Prime Number.

Typically when finding the factors of a number we are trying to find its lowest, most broken down, or prime factors. For example the number 100 could be broken down into 4 and 25. While these are factors of 100, each can still be broken down into smaller numbers. 4 can be broken down into 2 and 2. 25 can be broken down into 5 and 5. We can combine these to get the full list of the prime factors of 100 as 2, 2, 5, and 5.

Factors of 100:

A Root is a specific factor, when multiplied by itself a number of times (depending on what "root" it is), creates the number. For example 2 is the Square Root of 4, as well as the Cubed Root of 8.

√4 = 2
2 * 2 = 4

∛8 = 2
2 * 2 * 2 = 8

The symbol to denote finding the root of a number is the Radical Symbol. The number outside or above the "v" part of the radical indicates the degree of the root (i.e. how many of the root are multiplied together to get the number inside the radical). If the radical is left "empty" then it is convention that it represents the Square Root (i.e. 2).

More Examples:
√4 = 2
2 * 2 = 4

∛8 = 2
2 * 2 * 2 = 8

∜16 = 2
2 * 2 * 2 * 2 = 16

√25 = 5
5 * 5 = 25

√9 = 3
3 * 3 = 9

∛343 = 7
7 * 7 * 7 = 343

∜6561 = 9
9 * 9 * 9 * 9 = 6561

You may have noticed in the last example above, 9, is not a prime factor of 6561. While a root is often a lowest factor, or a prime number, it does not have to be. The square root of 100 for example is 10. In fact, a root doesn't even have to be a whole number or can also be an irrational number, such as √2 (≈1.414).

More Examples:
√100 = 10
10 * 10 = 100

√625 = 25
25 * 25 = 625

∛3375 = 15
15 * 15 * 15 = 3375

√2 ≈ 1.4142
1.4142 * 1.4142 ≈ 2

∛5 ≈ 1.7099759
1.7099759 * 1.7099759 * 1.7099759 ≈ 5

Fractional Exponents, i.e. "Roots"

Fractional Exponents, i.e. "Roots"
x^a/b = z^a
where
z^b = x
i.e. z is the "b^th-root" of x

Fractional exponents may seem complicated, but the idea behind them is actually quite simple. Lets start by thinking of a number's roots as a part or "fraction" of that number's factors. Note that without multiplying all of the roots together we do not come out with the whole number.

Roots as a Fraction of a Whole Number:

Here we can begin to see how the fractional exponent correlates with the term's root factors. The denominator tells us the degree of the root it is dividing its term into. That is, it tells us how many root factors it is breaking the number into. The numerator tells us how many of those root factors we are keeping to multiply together.

With this in mind we can see how each fractional exponent relates to the first three root degrees.

First Three Root Degrees as Fractional Exponents:

√x = x^1/2,
because we are breaking x into two root factors and only keeping one.

While the denominator determines how many root factors we are breaking the number into, the numerator affects how many of them we "keep" to multiply together. It is essentially adding an exponent to the result of finding the root.

Root Equivalents of Fractional Exponents:

x^1/2 = (√x)¹,
because we are breaking x into two root factors and only keeping one.

To write this relationship, lets define z as a root factor of x, which is of degree b. That is to say that z^b = x or that the b root of x is z. If we then raise x to a fractional exponent in the form of a/b, (x^a/b) we would get z^a.

Relationship of Roots and Fractional Exponents:

z = ^b√x, i.e. the b^th root of x

so that:
z^b = x

then:
x^a/b = z^a

We can further illustrate this relationship by showing the factors we "keep" out of the normal root factors:

Examples of Fractional Exponents:

16^3/4 = (∜16)³ = 8

Exponents of Zero

Summary:

x⁰ = 1

Any number raised to the zero^th power is one. This may seem odd since typically you think of zero of something as zero, but it becomes evident once you do out the math.

Division of exponents can be used to help us better understand this. When dividing two terms you subtract the exponents, canceling out some of the terms on both sides of the dividing line. Consider what is happening to cancel out these terms and why. They can't be reduced to zero because it would cause the remaining terms to be reduced down to zero as well when multiplied. The reason they cancel out is because you are dividing a number by itself, which is always one.

Division of Same Terms Cancel out:

Now consider the case of when you have an equal number of terms on either side of the dividing line. All terms would "cancel" each other out. Each and every term would be reduced to one, and one multiplied or divided by any number of ones will remain one.

Division of Same Terms Cancel out:

Exponents of One

Summary:

x¹ = x

Any term raised to an exponent of one is itself. This actually makes perfect sense as it is essentially saying you have one group of a number. It can also be shown mathematically as previously done for exponents of zero.