Use the properties of real numbers to rewrite and simplified each expression. State
which properties apply.
Let's begin by reviewing the properties of real numbers.
First we have the commutative property of addition and the commutative property
of multiplication. The commutative property of addition is when we change
the order of a sum which states a plus b equals b plus a. The community property
multiplication is when we change the order of multiplication, which dates a
times b equals b times a.
Next we have the associative property which deals with grouping. The associative
property of addition states a times the quantity b plus c equals the quantity a
+ b + c. When we change the grouping of a sum we are using the associative
property.
The associative property of multiplication states a times b c equals a b times c.
When we change the grouping of multiplication, we are using the
associative property.
Next is the distributive property which states a times the quantity b plus c
equals a times b plus a times c. The distributive property is often referred
to as multiplication across addition.
Next is the identity property of addition and of multiplication. The identity
property of addition states there exists a unique real number called the additive
identity, 0, such that for any real number a + 0 equals a. The identity property of
multiplication states there exists a unique real number called the
multiplicative identity, one, such that for any real number a, a times 1 equals a. And
then finally we have the inverse property of addition and the inverse
property of multiplication. The inverse property of addition states every real
number a has an additive inverse or opposite denoted negative a of the
opposite of a such that a plus negative a equals 0.
The inverse property of multiplication states every nonzero real number, a, has a
multiplicative inverse or reciprocal denoted one over a such that a times 1
over a
equals 1. So going back to our example, looking at expression one.
Let's begin simplifying by clearing parentheses by distributing the four.
Using the distributive property, distributing the four we have 4 times 7
plus 4 times 5.
Again this is distributive
property. The next have in simplifying is to determine the products.
4 times 7 equals 28. 4 times 5 equals 20. 28 plus 20 equals
48. For the next expression
notice notice how nine and negative nine are opposites and therefore let's change the
grouping of this sum. Let's group the nine plus negative nine instead of the three
plus 9. Changing the grouping of addition is associative property of addition.
The given expression is equal to 3 plus the quantity 9 plus negative 9.
This is the associative property of addition. Now let's find the sum of
the opposites inside the parentheses. 9 plus negative 9 is equal to 0. This is
the inverse property of addition. And three plus zero is equal to three. Adding
0 is the identity property of addition. The given expression simplifies three
Next we have eight minus the quantity 11 + 6. To begin simplifying let's clear the
parentheses. Because of the subtraction we clear the parentheses by distributing
negative 1. To make this more obvious
let's write the given expression as a sum. 8 minus the quantity 11 plus 6
is equivalent to the expression 8 plus negative 1 times the quantity 11 + 6. In
this form it's much more obvious to clear the parentheses we distribute
negative 1. Distributing we have eight plus negative 1 times eleven plus
negative 1 times 6. This is the distributive property. Now we simplify by
determining the products. We have eight plus negative 1 times eleven is
negative 11. We can write plus negative 11 or just minus 11.
Let's go ahead and write plus negative eleven. Plus negative 1 times six equals
negative 6.
So we can write plus negative 6 or minus 6. Let's right plus negative 6. Now we
simplify. 8 plus negative 11 is negative 3 negative. Negative 3 plus negative 6 equals
negative 9.
The given expression simplifies to negative 9.
Next we have three fourth times the product of seven ninths and four thirds.
Well three fourths and four thirds are reciprocals. Therefore let's change
the order of this product inside the parentheses using the commutative
property of multiplication. That way we will have three fourths and four thirds next
to each other.
The given expression equals three fourths times the quantity four thirds
times seven-ninths due to the community property of multiplication. Now let's
change the grouping of the product. Let's group three fourths and four thirds,
which are reciprocals. This is an example of the associative property of
multiplication. The expression is equal to the quantity three fourths times four
thirds times seven-ninths, due to the associative property of multiplication.
And now we'll determine the product of the reciprocals. Three fourths times four
thirds equals 1, which is the inverse property of multiplication.
From here we know one times seven ninths equals seven ninths but because multiplying by 1 is an
example of the identity property application
let's first change you order this multiplication using the community
property multiplication. Then we can use the identity property of multiplication.
So the given product 1 times seven ninths is equal to 7 9 times one due to the commutative
property of multiplication. Then seven ninths x 1 is equal to seven ninths. Multiplying
by 1 is an example of the identity property of multiplication. Of course, we
would not go through all these steps to simplify the given expression but this
is a nice way to demonstrate many of the properties of real numbers.
Let's look at our last example. We have 100 times the quantity 0.23 plus
negative 1.78. We will begin by clearing the brackets or grouping symbols by
distributing the 100. Therefore we use the distributive property the given
expression is equal to 100 times 0.23
plus 100 times negative 1 point 7 8. Again this is the distributive property
And now we simplify by determining the products. 100 times zero point two three equals 23.
We have 23 + 100 times a negative 1 point 7 8 equals negative 178.
Now we find sum. 23 plus negative 178 equals negative 155. The given expression
simplifies to negative 155.
I hope you found this helpful.