Exponents - What are they?
Exponents are a fundamental math tool that you learn in Pre-algebra, but a lot of people continue to struggle with them all the way through calculus! Which is totally understandable, because exponents, not to mention related operations like fractions and radicals, can be really tricky. But that’s all the more reason to really conquer them. You really want to leave Pre-Algebra with a completely solid foundation in exponents.
In this video you'll learn:
0:38 // What are exponents?
1:13 // Which number is the base?
3:04 // How to read exponents?
3:44 // How to read exponents with special names?
5:26 // Why are exponents used in math?
7:57 // How to simplify exponents?
8:17 // When exponents are 0?
9:51 // When the exponent is 1?
11:46 // When exponents are multiplied?
15:58 // When exponents are negative?
20:36 // When exponents are added?
24:10 // When exponents are subtracted?
30:25 // Where are exponents in order of operations (PEMDAS)?
32:23 // How to deal with negative bases?
35:59 // When exponents are fractions?
39:42 // A complicated example
There’s no difference between exponents and powers. So exponents are also known as powers, and putting an exponent on something is the same as raising that something to a power. Exponents are just a shorthand way of indicating repeated multiplication of the same thing by itself.
When you see an exponential function, you’ll see the larger number on the bottom, and then a little small number in the upper right-hand corner next to it. The big number on the bottom is called the “base”, and the little number in the corner is called the exponent. You’re always raising the base to the exponent.
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There are lots of different ways to read an exponent, but the basic template is “base to the power”. In other words, if you have 2^4, you could read that as “2 to the 4”, “2 to the power of 4”, “2 to the 4th”, “2 to the 4th power”, “2 raised to the 4”, “2 raised to the fourth power”, or “2 raised to the power of 4”. All of these are acceptable and correct, but the simplest way, and the way you’ll hear exponents most commonly read, is “2 to the 4th”.
Because we use them so often, we have special names for the exponents 2 and 3. For an exponent of 2, we used the word “squared”, and for an exponent of 3, we use the word “cubed”.
Exponents are important in math because they allow us to abbreviate something that would otherwise be really tedious to write. If we want to express in mathematics the product of x multiplied by itself 7 times, without exponents we’d only be able to write that as xxxxxxx, x multiplied by itself 7 times in a row. So we need a different way to express that value, and that’s what exponents are used for. Instead of writing out x multiplied by itself 7 times, we can write x^7.
The rules of exponents are the 0 rule, the 1 rule, the power rule for exponents, the negative exponent rule, the product rule, and the quotient rule. They are the rules you use in order to simplify exponent problems and solve problems with exponents. And when you’re simplifying exponent problems, you want to apply the rules in the order I just listed, because exponents have their own unique order of operations.
The 0 rule is what you use when the exponent is 0. The rule is that anything raised to the power of 0 is equal to 1. The only exception is 0^0, which is an indeterminate form.
The 1 rule is what you use when the exponent is 1. The rule is that anything raised to the power of 1 retains its value. So 2^1 is still just 2, and 7^1 is still just 7.
The power rule is what you use when you raise a power to a power. This is also the rule where you multiply the exponents together. So when you have (x^2)^3, the result is x^6.
The negative exponent rule is what you use when your exponent is negative. To make the exponent positive, just move it from the numerator to the denominator, or vice versa. In other words, move it to the other side of the fraction, and the exponent will change from negative to positive.
The product rule is what you use when you're multiplying two terms with like bases. This is the rule where you add the exponents together. So when you have (x^2)(x^3), the result is x^5.
The quotient rule is what you use when you're dividing two terms with like bases. This is the rule where you subtract the exponents. So when you have x^5/x^12, the result is 1/x^7, because the term always stays with the larger exponent.
In order of operations (PEMDAS), exponent rules are applied after you simplify everything inside the parentheses.
When you have negative bases, remember that the negative sign has to be inside the parentheses in order for the exponent to apply to it. So (-x)^2 is (-x)(-x)=x^2. But -x^2 is -(x^2)=-x^2.
When exponents are fractions, they can be converted to roots, or radicals.They can also be rewritten using power rule.