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Rules of Exponents

 

An exponential expression is composed of two parts. The base "carries" the exponent on its upper right corner.

 

  Location Description
  • Base
b^n where b is the base The number or variable that is being multiplied repeatedly in the expanded form.
  • Exponent
b^n where n is the exponent The value that specifies how many times the base will be multiplied by itself.

 

For instance, how would you write 2*2*2*2*2 in exponential notation?

The number 2 is the number being multiplied repeatedly and so it automatically becomes the base of the exponential expression. Notice that it is written five (5) times. This value specifies the number of occurrences of the base, thus, this must be the exponent.

 

appears 5 times; 2*2*2*2*2=2^5 where 2 is the base and 5 is the exponent Read as "2, raised to fifth power"

 

The base of an exponential expression can also be a letter or variable. Suppose we have x*x*x*x*x*x*x*x*x*x. Since the variable x is multiplying itself ten (10) times, we can write this in a compact form.

 

appears 10 times; x*x*x*x*x*x*x*x*x*x=x^10 where x is the base and 10 is the exponent Read as "x, raised to the tenth power"

 


Now, let's go over the five (5) basic rules or laws of exponents.

Rules of Exponents
Rules Formula Description
Rule 1: Zero Rule b^0=1 Any nonzero number raised to zero power equals 1. Examples
Rule 2: Negative Rule b^-n=(1/b^n) or (1/b^-n)=b^n Any number raised to a negative exponent is not allowed! Move the base with a negative exponent to the opposite side of the fraction, then make the exponent positive. Examples
Rule 3: Product Rule b^n*b^m=b^n+m When multiplying exponential expressions with the same base, copy the common base then add their exponents. Examples
Rule 4: Division Rule b^n/b^m=b^n-m When dividing exponential expressions with the same base, copy the common base then subtract the top exponent by the lower exponent. Examples
Rule 5: Power to a Power Rule (b^n)^m=b^nm When an exponential expression is raised to a power, copy the base, then multiply the inner and outer exponents. Examples

 


Rule 1: Zero Rule of Exponent (EXAMPLES)

  • Simplify the exponential expression 5^0.

We have a nonzero base of 5, and an exponent of zero. The zero rule of exponent can be directly applied here. Thus, 5^0=1.

  • Simplify the exponential expression (2x^2y)^0.

The base here is the entire expression inside the parenthesis, and the good thing is that it is being raised to the zero power. Caution, as long as variables x and y don't assume the value of zero, we can definitely apply the zero rule of exponent here as well. This gives us

(2x^2y)^0

 

  • Simplify the exponential expression [5(x^3)^0]/[15(y^3)^0].

Each of the expression inside the parenthesis with zero power found in both numerator and denominator will simply be replaced by 1. Make sure to reduce the fraction to its lowest term.

[5(x^3)^0]/[15(y^3)^0 = 5(1)/15(1) = 5/15 = 1/3

 


Rule 2: Negative Rule of Exponent (EXAMPLES)

  • Simplify the exponential expression 2^-4.

The base 2 has a negative exponent of -4. This can be fixed by moving it to the denominator, and switching the sign of the exponent to positive using the negative rule of exponent.

2^-4=1/2^4=1/16
  • Simplify the exponential expression 1/x^-7.

This time the base with a negative exponent is found in the denominator. Bring it up to the numerator while making the exponent positive.

1/x^-7 = x^7

 

  • Simplify the exponential expression x^-2/y^-3.

Both of the exponents in the numerator and denominator are negative. It should make sense to swap their locations along the fractional bar. The x-variable goes down, while the y-variables goes up! Make sure to change both their exponents to positive.

x^-2/y^-3 = y^3/x^2

 


Rule 3: Product Rule of Exponent (EXAMPLES)

 

  • Simplify the product of exponential expressions (x^6)(x^2).

We are multiplying two exponentials with the same base, x. The product allows us to combine them by copying the common base, and then adding their exponents.

 

(x^6)(x^2) = x^6+2 = x^8

 

  • Simplify the product of exponential expressions (2x^3y^9)(7x^2y^2).

Observe that each parenthesis contains a number, x-variable and y-variable. Make sure to multiply the terms of the same kind only. That is, multiply the numbers together, and multiply each kind of variable separately. To emphasize this step, we will group them first before applying the product rule.

(2x^3y^9)(7x^2y^2)=(2*7)(x^3*x^2)(y^9*y^2)=14(x^3+2)(y^9+2)=14x^5y^11

 

  • Simplify the product of exponential expressions (x^6y^-2)(x^-13y^2).

After we multiply the exponential expressions with the same base by adding their exponents, we arrive at having one variable with a negative exponent, and another with zero exponent.

Don't hesitate to apply the two previous rules learned, namely Rule 1 and Rule 2 , to further simplify this expression.

 

((x^6y^-2)(x^-13y^2)=(x^6*x^-13)(y^-2*y^2) = [x^6+(-13)][y^-2+2] = x^-7y^0 = 1/x^7; negative exponent rule; zero rule

 


Rule 4: Division Rule of Exponent (EXAMPLES)

  • Simplify the quotient of exponential expressions x^7/x^4.

The fractional bar implies that we are going to divide. It makes sense to apply the Division Rule of Exponent, that is, copy the common base in the numerator and denominator and subtract the top exponent by the lower exponent.

x^7/x^4=x^7-4=x^3

 

  • Simplify the exponential expressions x^2y^9/x^7y^5.

Comparing the expressions in the numerator and the denominator, I see that there are two common bases, x and y. Apply the division rule on each variable. After doing so, the x-variable will contain a negative exponent, therefore, use the negative rule of exponent to fix the problem.

x^2y^9/x^7y^5 = (x^2-7)*(y^9-5) = x^-5*y^4 = y^4/x^5

 

 

  • Simplify the exponential expressions x^-9y^3/x^-7y^8.

One way to simplify this is to ignore the negative exponents for now. Apply the division rule first, and see if negative exponents show up again. If that's the case, utilize the negative rule of exponent.

x^-9y^3/x^-7y^8 = x^-9-(-7)*y^3-8 = x^-9+7*y^-5 = x^-2y^-5 = 1/x^2y^5

 


Rule 5: Power to a Power Rule (EXAMPLES)

  • Simplify the exponential expression (x/5)^3.

This expression has an inner and outer exponent. The Power to a Power Rule allows us to copy the base and multiply the exponents.

(x/5)^3 = x^5*3 = X^15

 

  • Simplify the exponential expression (2^3x^7y^6)^2.

This problem is very similar to the previous one. The only difference is that there are three (3) inner exponents. We just need to distribute the outer exponent to each of the inner exponent.

(2^3x^7y^6)^2 = (2^3*2)(x^7*2)(y^6*2) = 2^6x^14y^12 = 64x^14y^12

 

  • Simplify the exponential expression (3x^4/y^6)3.

Although we have a fractional expression inside the parenthesis, the Power to a Power Rule still holds! Distribute the outer exponent to all inner exponents.

(3x^4/y^6)3 = (3^3x^12)/y^18 = 27x^12/y^18

 


 

Practice Problems with Answers
Worksheet 1 Worksheet 2

 

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