What Are Exponents?
Learning about exponents is important as you move on to more advanced math. You no longer need to struggle because this blog is your guide to learning about exponents! While this seems like a tough topic right now, continue reading!
What are Exponents in Math?
An exponent is a number written to the top right of another number. It tells us how many times a number is multiplied by itself.
For example, take 5^2. This is equal to 25 because the exponent is 2, so that tells us to multiply the base (5) by itself 2 times (5 x 5). Using exponents comes in handy when expressing large numbers. They can be positive, negative, zero, and even rational!
With all these different ways to use exponents, there are a couple of rules you need to keep in mind! You call these the rules of exponents. If you were to solve a question like 6^4-6^9, there’s an easy way to find the answer! You would use the rules of exponents.
Exponents 101: What are the Rules for Exponents?
There are many exponent rules, so we’ll be breaking down quite a few of them one by one! Let’s start with adding and subtracting!
When it comes to addition and subtraction, it’s about combining like terms. What does “like terms” mean? It’s the common variables and exponents that you can combine or separate. For example, if we were asked to subtract 6w from 8w, we would get 2w.
If we were asked to subtract 6w from 8a, you wouldn’t be able to do that because the variables are different assuming w≠a. You could also expand 8w in many ways like 4w+4w or 2w+2w+2w+2w.
To look at multiplication, we have the Product Rule. The rule is x^a*x^b = x^a+b. When multiplying two of the same bases with different powers, you add the powers together. To clarify, the bases are the variables, not the coefficients! To give an example, if we were to take h^2*h^6, that equals h^8 because h^2*h^6 equals h^2+6 which equals h^8.
For the Power Rule, the rule is (x^a)b = x^ab. Since we’re raising x^a to the b, we would be multiplying the powers together instead of adding. For example, (x^3)^2 is x^6 because (x^3)^2 is equal to x^3*2 which is x^6. Something else to keep in mind is that (ab)^c = a^c*b^c. So if we have (5k^4)^2, the simplified version would be (5^2)(k^4)^2 which is (25)(k^8), simlifying to 25k^8.
Now, let’s look at the Quotient Rule which is x^a/x^b = x^a-b. When we’re using division, we’re subtracting the exponents. Don’t forget that the bases must be the same! This is like the Product Rule because when we’re using multiplication, we’re adding the powers! To illustrate, let’s consider 4x^9/2x^5. This is 2x^4 because 4x^9/2x^5 is (4/2)x^9-5 which is 2x^4.
Moving on, we have the Negative Exponent Rule which can be described as x^-a = 1/x^a. So if we were to simplify -5x^-2 it would be -5/x^2. This is a tricky example, but what’s going on is that the power only applies to x, not the -5. -5x^-2 is equal to -5*x^-2 which is -5*1/x^2 when you use the Negative Exponent Rule on x^-2. From here, we can use multiplication and multiply -5 with 1 because -5 is -5/1 and that times 1/x^2 is -5/x^2.
Last but not least, we’re going to talk about the Zero Exponent Rule. This is one of the easier rules to remember because ^x0=1.
For example, 6^0 equals 1. To move on to something a bit more advanced, let’s try to simplify 7x^0. The answer is 7 instead of 0 because you must remember that the power of x applies to x, not 7. This would mean 7x^0 is 7*1 because 7x^0 = 7*x^0 and we know that x^0 equals 1.
You have to be careful with this rule though because this rule doesn’t always work! For instance, we can’t raise 0 to a negative exponent or zero exponents! This is true since if we have a number like 0^-5, that would be 1/0^5. Since 0^5 is 0, we get 1/0 which is undefined. 0^0 also doesn’t work because 0 raised to any positive power is 0, so 0 to the power 0 should be 0.
But at the same time, any positive number raised to the power 0 equals 1, so 0 to the power 0 should equal 1. 0^0 can’t be 1 and 0, so to avoid breaking math, we say 0^0 is undefined.
Practice Exponent Problems
Adding and subtracting monomials
1. Try simplifying 9x^2y-10x^2y.
Answer: A monomial is one term. For example, 7x^3y^2z is a monomial because 7 is the coefficient while x, y, and z are variables. 3 and 2 are the exponents.
2. Try simplifying 5y-2y.
Answer: 3y
Product rule
3. Try simplifying (-2a^2b)*(7a^3b)
Answer: -14a^5b^2 because 2*7=14. a^2*a^3 is a^2+3 which is a^5. The b variables don't have a power written, but when that happens it's safe to assume that the power is 1 because b^1 is b. b^1*b^1 is b^2. When we put -14, a^5, and b^2 together, we get -14a^5b^2.
4. Try simplifying y^5*y^4
Answer: y^9
Power Rule
5. Try simplifying (x^3)^2.
Answer: x^6
Steps:
1. (x^3)^2
2. x^3*2
3. x^6
6. Try simplifying (-2m5)2*m3
Answer: 4m^13
Steps:
1. (-2m^5)2*m^3
2. (-2^2m^5*2)*m^3
3. (4m^10)*m^3
Now we can use the product rule.
4. 4m^10+3
5. 4m^13
Quotient Rule
7. Try simplifying 27x^5/42x.
Answer: 9x^4/14
Steps:
1. (27x^5)/(42x)
2. (9x^5)/(14x)
3. (9x^5-1)/(14)
4. 9x^4/14
8. Try simplifying (y^2)^2/y^4.
Answer: 1
Steps:
1. (y^2)^2/y^4
2. (y^2*2)/y^4
3. y^4/y^4
4. 1
Negative Exponent rule
9. Try simplifying -10y^-2.
Answer: -10/y^2
Steps:
1. -10y^-2
2. -10*y^-2
3. -10*1/y^2
4. -10/y^2
10. Try simplifying 4k^2/8k^5.
Answer: 1/2k^3
Steps:
1. 4k^2/8k^5
2. 1k^2/2k^5
3. 1k^2-5/2
4. 1k^-3/2
5. 1/2k^3
Zero exponent rule
11. Try simplifying 10d^0.
Answer: 10
Steps:
1. 10d^0
2. 10*d^0
3. 10*1
4. 10
12. Try simplifying (w^4)^2/w^8.
Answer: 1
Steps:
1. (w^4)^2/w^8
2. w^8/w^8
3. w^8-8
4. w^0
5. 1
Final Thoughts on Exponents
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