SIMPLIFYING EXPRESSIONS

Objectives:  To recall how to simplify expressions using associative, commutative and distributive rules of arithmetic.

Recall 1Commutative laws for addition and multiplication.
These   laws tell us that the order in which we add or multiply   numbers is not important.

Adding 5 and 6 is the same as adding 6 and 5.  The order in which we add any two numbers is not important, as the equalities below illustrate.

5 + 6 = 6 + 5,
3 + 7 = 7 + 3,
x + y = y + x.

The law that permits us to flip numbers around an addition sign is called the commutative law for addition.  It holds for any real numbers x and y, fractions or decimals, as well as for integers.  In other words

or

2.343 + 5.437 = 5.437 + 2.343.

Multiplying 5 and 6 is the same as multiplying 6 and 5.  The order in which we multiply any two numbers is not important, For example:

5 *6 = 6 * 5,
3 * 7 = 7 * 3,

and, more generally,

x * y = y * x.

The law that permits us to flip numbers around a multiplication sign is called the commutative law for multiplication.  It holds for any real numbers x and y, fractions or decimals, as well as for integers.  In other words

2.343 * 5.437 = 5.437 * 2.343.

Recall 2Associative laws for addition and multiplication.
These   laws tell us that we may move parentheses around.

Adding 5 and 6, then adding 7, is the same as adding 5, then adding the sum of 6 and 7.  This works for any three numbers, as the equalities below illustrate.

(5 + 6) + 7 = 5 + (6 + 7),
(3 + 7) + 9 = 3 + (7 + 9),

or, more generally,

(x + y) + z = x + (y + z)

The law that permits us to move parentheses around in this manner is called the associative law for addition.  It holds for x and y and z  fractions or decimals, as well as for integers.  In other words

(2.343 + 5.437) + 3.546 = 2.343+ (5.437 + 3.546)

Multiplying 5 and 6, then multiplying the result by 7, is the same as multiplying 5 by the product of 6 and 7.  Moreover:

(5 * 6) * 7= 5 * (6 * 7),
(3 * 7) * 9 = 3 * (7 * 9),

and, more generally,

(x * y) * z = x * (y * z)

This law is called the associative law for multiplication.  It holds for x and y and z  fractions or decimals.  In other words

(2.343 * 5.437) * 3.546 = 2.343.* (5.437 * 3.546).

Recall 3Distributive laws for multiplication over addition.
This   law will be very useful in simplifying complicated   algebraic expressions.

Multiply 5 and (6 + 7). You get 65. Now add 5 x 6 to 5 x 7.  You again get 65.  In other words

5 * (6 + 7) = 5 * 6 + 5 * 7

Again, we should notice that the 5, 6 and 7 may be replaced by any three numbers.  For example, for 3, 7 and 9 we have:

3 * (7 + 9) = 3 *7 + 3 * 9

More generally, we find that

x * (y + z) = x* y + x * z

The law that permits us to write x * (y + z) as x * y + x * z is called the distributive law for multiplication over addition. It holds for x, y, and z fractions or decimals.  In other words

2.343 *( 5.437 + 3.546) = 2.343.* 5.437 + 2.343 * 3.546

The examples below demonstrate how the three laws just described (the commutative, associative and distributive laws) allow us to simplify complicated expressions.

Examples

Example 1.  Simplify the expression   3*(5 + 3*(x + 9) + y).

Solution: Work with the middle term inside the parentheses first, 3*(x + 9). The distributive law says that this term must be the same as
3*x + 3*9,
or
3*x + 27.

Replacing 3*(x + 9) by 3*x + 27, we find that the expression may be written as
3*(5 + 3*x + 27 + y).

The commutative law says that we may flip the 3*x and the 27, to get an expression that looks like:

3*(5 + 27 + 3*x + y).

The associative law allows us to rewrite the expression as

3*((5 + 27) + 3*x + y),

or as

3*(32 + 3*x + y).

Finally, we may use the distributive law once again to distribute the 3 over the addition to get the final simplified form

3*32 + 3*3 x + 3*y,
or
96 + 9*x + 3*y.

Note
The laws discussed above work equally as well for negative numbers, though we sometimes must interpret the results more carefully.  For example,

(3 - 5) is different from (5 - 3).

However, (x - y) means x + (-y); so, if the first number is 3 and the second is -5 then indeed the commutative law for addition does work and

3 + (-5) = (-5)+3.

Example 2.   Simplify the expression    3 x ((x - y) + 5*(x + y)).

Solution:   Always work with the innermost parentheses first.  In this case we work with
(x - y) + 5*(x + y).

It simplifies to become
(x - y) + 5*x + 5*y, (by the distributive law)
then
x - y + 5*x + 5*y, (by the associative law)
then
x + 5*x + (-y) + 5*y, (by the commutative law)
then
(x + 5*x) + (-y + 5*y) (by the associative law)
and finally
6*x + 4*y (by the associative law)

So, the expression inside the innermost parentheses is 6*x + 4*y.  Recall that the original expression was 3*x*((x - y) + 5*(x + y)), which may now be represented as

3*x*(6*x + 4*y)
This simplifies to
(3*x)*(6*x) + (3*x)*(4*y) (by the distributive law)

(3*6)*(x*x) + (3*4)*(x*y) (by the associative law)

18*x^2 + 12*x*y (by multiplication rules)

Example 3  Show that .

Solution: (x + y)^2 means (x + y)*(x + y).  Use the distributive law once to get

(x + y)*x + (x + y)*y,

the commutative law for multiplication to get

x*(x + y) + y*(x + y),

and the distributive law again in each of the two terms to get

x*x + x*y + y*x + y*y,
or
x^2 + 2*x*y + y^2.

Example 4 Show that

Solution: Let y = 1 in Example 3.

Warning 1:  Always check your answers. In algebra, it is usually simple  to check an answer.  The expression will become a real number, if you  declare the unknowns to be specific real numbers.  So, pick values for  the unknowns, and you should get the same unknown for the original  expression as you do for the simplified expression.   The first  example demonstrated that 3*(5 + 3*(x + 9) + y) can be rewritten as      96 + 9*x + 3*y.  This means that

(equality 1)  3*(5 + 3*(x + 9) + y).= 96 + 9*x + 3*y.

If this last equality is true, then it must be true for any numbers that x and y stand for. In particular, it must be true for x = 1 and y = 1.

If we let x = 1 and y = 1, then this equation becomes
3*(5 + 3*(1 + 9) + 1).= 96 + 9*1 + 3*1.

Since both the left and right sides are equal to 108, the equality 1 stands a good chance of being correct.

Note:   Checking, by substituting simple numbers for x and y, does   not guarantee that Equality 5.1 is correct, as it may have been   a coincidence that the two expressions were equal for those   specific choices for x and y; however, such simple choices of x   and y usually would determine an error, had one occurred.    So, it is important to get into the habit of checking for errors.

Warning 2:  Your biggest frustrations will occur when you work out an  exercise with haste and carelessness.  It does not take many mistakes to  shake your self confidence; that is a human condition.  Studies show that  most students find math difficult because of a lack of self confidence  rather than a lack of ability.  So take your time and check your results.

You are now ready to carefully do the workout exercises.

EXERCISES (Workout 5)

Evaluate or simplify each of the following expressions:

1. 2*(5 + 4) - 3*(2 + 4)

2. 3*(4*(5 + 3) + (2 + 3))

3.  2*(2*(2 + 2) + 2*(2 + 2))

4. 2*(x + 4) - 3*(x + 4)

5. 3*(4*(x + 3) + (2 + x))

6. 2*(2*(x + 2) + 2*(x + 2))

7. 2*(5 - 4) - 3*(2 - 4)

8. -3*(4*(-5 + 3) + (2 - 3))

9. 3*(x*(5 - 3) + (x - 3))

10. 2*(x*(2 + x) + 2*(x + 2))

11. 2*(5 + z) - 3*(2 + z)

12. x*(5 - 4) - x*(2 + 4)

13. y*(4*(x + 3) + (y + x))

14. -3*y*(x - 3) - (-x*(y + 4))

15. -4*x*(z + y) + (x*(x - y))

16. -4*a*((c + b) + a*(a - b))

17. 3*((h + k - m) + 3*(h - k))

18. 3*((x^2 - 2*y + z) - (x - x*y))

Checking for common errors:

1. Suppose that you are given the expression x*(x + 2) to simplify, and that  you used the distributive law incorrectly to answer x^2 + 2.  (It is easy to  see how this typical mistake can happen; you may forget to multiply the  2 by the x.)   Next, suppose that you wish to check your result, which  means checking that

Equality 5.2  x*(x + 2) = x^2 + 2.

You choose to check the equality at x = 1.  So you check that

1*(1 + 2) = 1^2 + 2.

You calculate that the right side and the left side are both equal to 3.   You may feel that your answer is correct, but it isn't.  If you want a  better test for the accuracy of your answer, it is wise to plug in another  simple value for x, say x = 0.  Now, if you evaluate both sides of  Equality 5.2 at x = 0, you find that the left side is 0, while the right is 2.    Therefore, your answer has failed the test.

The lesson to be learned is that if you are not sure of an answer, check  it at at least one value of each of the unknowns.  It is generally unlikely  that the beginning expression and the answer expression will both  accidentally have the same value when evaluated at a single value of  each of the unknowns; however, it is even more unlikely that checking  at two values of each of the unknowns will accidentally give the same  value.

2. A majority of the common errors that you will encounter in performing  the exercises of the Workout 5 will be careless ones.  Here are  several that you may have made in the last workout.  Can you find the  errors in the solutions to Workout 5, Exercises below?   (Note:  these are samples from students who volunteered to work through  the first draft of Day Algebra Workout.)

Incorrect solution
to Exercise 6
2*(2*(x + 2) + 2*(x + 2))
2*(2*x + 4 + 2*x + 2)
2*(6 + 4*x)
12 + 8*x

Incorrect solution
to Exercise 9
3*(x*(5 - 3) + (x - 3))
3*(5*x - 9 + x - 3)
3*(6*x -12)
18*x - 36

Incorrect solution

to Exercise 14
-3*y*(x - 3) - (-x*(y + 4))
-3*y*(x - 3) - (-x*y - 4*x)
-3*y*(x - 3 + x*y + 4*x)
-3*x*y - 3*y - 3*x*y2 - 12*x*y
- 15*x*y - 3*y - 3*x*y2.

Incorrect solution
to Exercise 18

Here is a typical error, 3*((x^2 - 2*y + z) - (x - x*y))
where the student   3*(x^2 - 2*y + z - x + x*y)
carelessly copied 3*x - 6*y + 6 - 3*x + 3*x*y.

There are two errors in this last solution -- one is the result of not writing clearly enough to distinguish between a z and a 2.

1. 0     10. 2*x^2 + 8*x + 8

2. 111     11. 4 - z

3. 32     12. -5*x

4. -x - 4     13. 5*x*y + 12*y + y^2

5. 15*x + 42     14. -2*x*y + 9*y + 4*x

6. 8*x + 16     15. x^2 - 4*x*z - 5*x*y

7. 8     16. 4*a^2*b - 4*a*c - 4*a*b - 4*a^3

8. 27     17. 12*h - 6*k - 3*m

9. 9*x - 9     18. 3*x^2 - 6*y + 3*z - 3*x + 3*x*y