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
| Property | Of Addition | Of 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 \(a\) | \(a,a\ne 0\)
|
| For any real number a, | \(a+\text{(}\text{−}\mathit{\text{a}}\text{)}\phantom{\rule{0.2em}{0ex}}=0\) | \(a·1a=1\) |
| Distributive Property
|
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| 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 \(a,a\ne 0\) | \(\frac{0}{a}=0\)
|
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This lesson is part of:
Properties of Real Numbers
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