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| 13 Theory slides |
| 11 Exercises - Grade E - A |
| Each lesson is meant to take 1-2 classroom sessions |
Here are a few recommended readings before getting started with this lesson.
Ignacio got a part-time job at a restaurant. His first task is to peel some garlic. Ms. Ley, the restaurant owner, tells him to peel all the heads of garlic in a crate.
Commutative Property of Multiplication
Associative Property of Multiplication
a^m*a^n=a^(m+n)
Commutative Property of Multiplication
a^m*a^n=a^(m+n)
Multiply
Ignacio is helping Ms. Ley make hand-pulled noodles.
2*2 = 2^2 Ms. Ley told Ignacio to fold and stretch the noodles four more times. 2* 2 * 2 * 2 = 2^4 Notice that the power corresponds to the number of times Ignacio has stretched and folded, or doubled, the noodles. Since the entire amount of noodles is being doubled each time the dough is folded, the two monomials are multiplied. 2^2 * 2^4 The Product of Powers Property can be used to add the powers. 2^2 * 2^4 = 2^6 This numeric expression represents that Ignacio folding and stretching the noodles six times.
2^6 Multiply 2 by itself six times to evaluate this numerical expression. 2^6 &= 2*2*2*2*2*2 &= 64 This means that Ignacio made 64 noodles by folding and stretching the dough six times.
To encourage Ignacio to keep doing a great job, Ms. Ley gives him a bonus each time he gets a compliment from a customer.
This bonus is a small percent increase of his normal total wage for the day. Ignacio does not know how much this percentage is, so he thinks of it as x. This means that if he gets three compliments, his payment is multiplied by x three times. x* x* x = x^3 Ignacio gets two compliments from Table 1, four compliments from Table 2, and one compliment from Table 3. He writes an expression to show his compliment bonus for the day. x^2* x^4* x
x^2* x^4* x ⇓ x^2 * x^4 * x^1 Use the Product of Powers Property to multiply the monomials by adding the exponents. x^2 * x^4 * x^1 &= x^(2+4+1) &= x^7 This means that Ignacio will get a bonus of x^7. The degree of this monomial is 7. Notice that this is also the total number of compliments Ignacio got from all the tables. This is because the percentage bonus is applied each time he receives the a compliment.
Associative Property of Multiplication
Commutative Property of Multiplication
a* a=a^2
Associative Property of Multiplication
a^m*a^n=a^(m+n)
Multiply
Everyone has bad days and today was no exception. In addition to the four compliments from the customers, Ignacio also got two complaints.
Unfortunately, Ms. Ley is very strict and told Ignacio that for each complaint she receives about him, she would cancel out one of the compliment bonuses he gets. This situation can be represented with the following division of monomials. x^4/x^2
x^4/x^2 The Quotient of Powers Property can be used to divide the monomials because the bases are the same. Do this by subtracting the exponent of the denominator from the exponent of the numerator. x^4/x^2 &= x^(4-2) &= x^2 This means that Ignacio will only get the bonus from two compliments. Be more careful the next time, Ignacio!
x^4y^3/x^2y^2 = x^4/x^2 * y^3/y^2 Next, use the Quotient of Powers Property for each division. x^4/x^2 * y^3/y^2 &= x^(4-2) * y^(3-2) &= x^2 * y^1 Any number raised to the power of 1 is equal to itself. x^2 * y^1 = x^2 * y Finally, multiply the monomials. x^2 * y = x^2y The degree of the resulting monomial is the sum of the exponents of its variable factors. 2+1=3 This means that the degree of the monomial is 3. Ignacio got a total of three bonuses. Not too bad, but Ignacio is determined to do better!
It is time to make dumplings! Ignacio wants to help Ms. Ley with the dough. He plans to roll it out and cut it into small squares.
The following monomial shows how many squares Ignacio will have after cutting the dough sheet into an x-by-x grid. x^2
Ignacio was asked to peel all the heads of garlic in the crate.
Associative Property of Multiplication
Commutative Property of Multiplication
Associative Property of Multiplication
a^m*a^n=a^(m+n)
Multiply
Write each expression as a monomial using exponents.
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
Write each expression as a monomial using exponents.
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.
Multiply the monomials.
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
Divide the monomials.
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
Multiply and divide the monomials as required. Write the resulting expression in its simplest form.
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.
Write the degree of the given monomial.
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.