Mathematics » Properties of Real Numbers » Properties of Identity, Inverses, and Zero

Simplifying Expressions using the Properties of Identities, Inverses, and Zero

Simplifying Expressions using the Properties of Identities, Inverses, and Zero

We will now practice using the properties of identities, inverses, and zero to simplify expressions.

Example

Simplify: \(3x+15-3x.\)

Solution

 \(3x+15-3x\)
Notice the additive inverses, \(3x\) and \(-3x\).\(0+15\)
Add.\(15\)

Example

Simplify: \(4\left(0.25q\right).\)

Solution

 \(4\left(0.25q\right)\)
Regroup, using the associative property.\(\left[4\left(0.25\right)\right]q\)
Multiply.\(1.00q\)
Simplify; 1 is the multiplicative identity.\(q\)

Example

Simplify: \(\frac{0}{n+5}\), where \(n\ne -5\).

Solution

 \(\frac{0}{n+5}\)
Zero divided by any real number except itself is zero.\(0\)

Example

Simplify: \(\frac{10-3p}{0}.\)

Solution

 \(\frac{10-3p}{0}\)
Division by zero is undefined.undefined

Example

Simplify: \(\frac{3}{4}·\frac{4}{3}\left(6x+12\right).\)

Solution

We cannot combine the terms in parentheses, so we multiply the two fractions first.

 \(\frac{3}{4}·\frac{4}{3}\left(6x+12\right)\)
Multiply; the product of reciprocals is 1.\(1\left(6x+12\right)\)
Simplify by recognizing the multiplicative identity.\(6x+12\)

All the properties of real numbers we have used in this tutorial are summarized in the table below.

Properties of Real Numbers

PropertyOf AdditionOf Multiplication
Commutative Property  
If a and b are real numbers then…\(a+b=b+a\)\(a·b=b·a\)
Associative Property  
If a, b, and c are real numbers then…\(\left(a+b\right)+c=a+\left(b+c\right)\)\(\left(a·b\right)·c=a·\left(b·c\right)\)
Identity Property\(0\) is the additive identity\(1\) is the multiplicative identity
For any real number a,\(\begin{array}{l}a+0=a\\ 0+a=a\end{array}\)\(\begin{array}{l}a·1=a\\ 1·a=a\end{array}\)
Inverse Property\(-\mathit{\text{a}}\)is the additive inverse of \(a\)\(a,a\ne 0\)

 

\(1\mathit{\text{a}}\) is the multiplicative inverse of \(a\)

For any real number a,\(a+\text{(}\text{−}\mathit{\text{a}}\text{)}\phantom{\rule{0.2em}{0ex}}=0\)\(a·1a=1\)
Distributive Property

 

\(\phantom{\rule{10em}{0ex}}\)If \(a,b,c\) are real numbers, then \(a\left(b+c\right)=ab+ac\)

Properties of Zero  
For any real number a,

 
\(\begin{array}{l}a\cdot 0=0\\ 0\cdot a=0\end{array}\) 
For any real number \(a,a\ne 0\)\(\frac{0}{a}=0\)

 

\(\)\(\frac{a}{0}\) is undefined

 

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