In this video we cover the basic rules for exponents, the product rule for exponents, the quotient rule for exponents, the power rule for exponents, negative exponents, and an exponent of zero.

Transcript/notes
Exponents rules basic
In the expression a raised to the 3, a is the base and 3 is the exponent. This can also be read as a to the 3rd power. And what this means is that we have 3 factors of a, so, we need to multiply a times itself 3 times, which is a times a times a.

For example, if we had 6 raised to the 3, it would be 6 times 6, times 6. This can be a bit confusing, so, you can break it down, 6 times 6 is 36, and now we have 36 times 6, which calculates out to 216. So, 6 raised to the 3 equals 216.

As another quick example, 3 raised to the 4. This would be 3 times itself 4 times, 3 times 3, times 3, times 3, which calculates out to 81.
One note, any number without an exponent has an exponent of 1, and all of the following rules only apply to exponents that have the same base value. For instance, 2 raised to the 3 and 5 raised to the 3 have different base values, so you could not apply any of the following rules.

Product rule
Now for a couple of rules for exponents, starting with the product rule for exponents. a raised to the m times a raised to the n equals a raised to the m plus n. So, we just add the exponents together. For instance, a raised to the 3 times a raised to the 4. Using the product rule, we just add the exponents together to get a raised to the 7 as the final answer. If we write this out, we have a times a, times a for a raised to the 3, times a times a, times a, times a, for a raised to the 4. So, we have a times itself 7 times, which is a raised to the 7, same answer as before.

Quotient rule
The quotient rule for exponents is a raised to the m divided by a raised to the n equals a raised to the m minus n. So, for division, we subtract the exponents. For instance, a raised to the 4 divided by a raised a raised to the 2. Using the quotient rule, we subtract the exponent in the bottom from the exponent on top, so, 4 minus 2 is 2, and we have a final answer of a raised to the 2. If we write this out, we have a times a, times a, times a for a raised to the 4 on the top, and a times a for a raised to the 2 on the bottom. We can cancel 2 of these out and we are left with a raised to the 2 as the answer, same as before.

Power rule
The power for exponents is the quantity a raised to the m, raised to the n equals a raised to the m times n. So, in this case, we multiply the exponents. For instance, the quantity 4 raised to the 2, raised to the 3. Using the power rule, we multiply the exponents, 2 times 3, which equals 6, so the final answer is 4 raised to the 6. If we write this out, 4 raised to the 2 is 4 times 4, and we have this quantity of 4 times 4 raised to the 3, so, the quantity 4 times 4, times the quantity 4 times 4, times the quantity 4 times 4. As you can see there are 6 4’s here, so 4 raised to the 6, same answer as before.

Negative exponents
The negative exponent rule is a raised to the negative n equals 1 over a raised to the n. So, in this case, we put the number or quantity being raised to the negative power in the denominator and the exponent becomes positive. For instance, 7 raised to the negative 2 is 1 over 7 raised to the positive 2. The quantity 1 minus y raised to the negative 4 equals 1 over the quantity 1 minus y raised to the positive 4.

A raised to the zero
A number raised to zero, such as a raised to the zero, is equal to 1. For instance, 3 raised to the zero equals 1. We can use the quotient rule to show this. As an example, 4 raised to the 4 divided by 4 raised to the 4. The quotient rule tells us to subtract the bottom exponent from the top exponent, 4 minus 4 is zero, and we have 4 to the zero as our answer, which is equal to one. If we write this out, we have 4 times 4, times 4, times 4 as the numerator and 4 times 4, times 4, times 4 as the denominator. All of these 4’s will cancel out, and we have 1 as the answer, same as before.

Timestamps
0:00 Basic overview of exponents
0:45 Any number without an exponent
1:03 Product rule of exponents
1:41 Quotient rule of exponents
2:23 Power rule of exponents
3:07 Negative exponents rule
3:33 Number raised to 0 (zero)

свернуть