Multiply and Divide Monomials

Let's take a look at the given expression. a * a * a * a * a * a This expression has a repeated factor a. It is multiplied by itself six times. Recall that a monomial can be written by taking the repeated factor as a base and the number of times that the factor is multiplied by itself as an exponent. This means that we can write the given expression as a monomial as a to the power of 6. a * a * a * a * a * a = a^6

This time the given expression has a repeated factor b. It is multiplied by itself three times. b * b * b This corresponds to b raised to the power of 3. b * b * b = b^3

The last expression is a value c multiplied by itself four times. c * c * c * c This corresponds to c raised to the power of 4. c * c * c * c = c^4

Let's take a look at the given expression. a * a * b * b * a * a Notice that this expression has two repeated factors, a and b. The factor a is repeated four times and the factor b is repeated two times. This means that we can write the given expression as a monomial as the product of the variable a raised to the power of 4 and the variable b raised to the power of 2. a * a * b * b * a * a = a^4b^2 Because of the Commutative Property of Multiplication, b^2a^4 is also a possible answer.

Let's take a look at the next expression. x * y * x * y This expression has x two times as a factor and y two times as a factor. This means that we can write it as a monomial just as we did on Part A. x * y * x * y = x^2y^2 Please note that y^2x^2 is a possible answer as well.

Let's take a look at the given expression. p^4 * p^7 The Product of Powers Property tells us that when we multiply monomials with the same non-zero base, we can add their powers. In our expression, we have two monomials with the base p. Let's use the Product of Powers Property to multiply them by adding the powers of our monomials. p^4 * p^7 &= p^(4+ 7) &= p^(11)

Let's take a look at the next expression. x^8 * x When a variable has no power written over it, the power is assumed to be 1. x = x^1 We can rewrite the given expression by using this information. x^8 * x^1 Let's now add the powers of the monomials x^8 and x^1 by the Product of Powers Property. x^8 * x^1 &= x^(8+ 1) &= x^9

Let's take a look at the last expression. z^2 * z^2 * z^4 This time we have three monomials. The base of all three monomials is z, so we can use the Products of Powers Property to multiply them. We just need to add the three powers! z^2 * z^2 * z^4 &= z^(2+2+4) &= z^8

Let's take a look at the given expression. p^8/p^4 The Quotient of Powers Property tells us that when we divide monomials with the same non-zero base, we subtract their powers. In our expression, we have two monomials each with the base p. Let's use the Quotient of Powers Property to divide them. We do this by subtracting the power of the denominator from the power of the numerator. p^8/p^4 &= p^(8- 4) &= p^4

Let's take a look at the next expression. q^5/q^4 In this expression, we have the division of two monomials with the base q. We will once again subtract the power of the denominator from the power of the numerator by using the Quotient of Powers Property. q^5/q^4 &= q^(5- 4) &= q^1 Any number raised to the power of 1 is equal to itself. q^1 = q Therefore, we can also express our result as q. q^5/q^4 = q

Let's take a look at the given expression. x^6 * x^3/x^5 Let's begin by multiplying the monomials in the numerator. Notice that they have the same base x. This means that we can use the Product of Powers Property to multiply them by adding the powers of the monomials. x^6 * x^3/x^5 &= x^(6+ 3)/x^5 [0.75em] &= x^9/x^5 The resulting monomials have the same base x. Now we can use the Quotient of Powers Property to divide them by subtracting the power of the denominator from the power of the numerator. x^9/x^5 &= x^(9- 5) [0.5em] &= x^4 The result is x^4.

Let's take a look at the next expression. z^9/z^2* z We can start by multiplying the monomials in the denominator. Notice that they have the same base z. Let's use the Product of Powers Property to multiply them by adding the powers of the monomials. Do not forget that if no power is shown, it is assumed to be 1. z^9/z^2* z &= z^9/z^(2+ 1) [o.75em] &= z^9/z^3 The resulting monomials have the same base z. Let's use the Quotient of Powers Property again to subtract the power of the denominator from the power of the numerator. z^9/z^3 &= z^(9- 3) [0.5em] &= z^6 The result is z^6.

Let's take a look at the given monomial. x^3 The power of this monomial is 3. This means that x is being multiplied by itself 3 times. x^3 = x * x * x Since there are 3 factors present in the above monomial, its degree is 3.

Let's take a look at the next monomial. x^2yz^4 This monomial has three variables. The x-variable is present as a factor 2 times, the y-variable just 1 time, and the z-variable 4 times. x^2y^1z^4 = x * x * y * z * z * z * z Add the number of times each variable is being used as a factor. The result will give us the degree of the monomial. 2+ 1+ 4 = 7 Another way of finding the degree of the monomial is calculating the sum of the powers of the variables. Keep in mind that when a variable has no power written over it, it is assumed that its power is 1. x^2yz^4 = x^2 y^1 z^4 ⇓ 2+ 1+ 4 = 7 This means that the degree of the given monomial is 7.