3.02 Simplify numerical expressions
Simplify numerical expressions
Recall the order of operations:
Evaluate operations inside grouping symbols.
Grouping symbols may include parentheses (\text{ }), brackets [ \text{ }], absolute value bars | \text{ }|, square root symbols \sqrt{\text{ }}, and fraction bars \frac{⬚}{⬚}.
Evaluate any exponents.
Evaluate any multiplication or division, reading from left to right.
Evaluate any addition or subtraction, reading from left to right.
Consider and check the sign of the final answer, especially if negative values were involved in the calculation.
The order of operations we used for integers extends to all rational numbers.
Here are some of the properties of real numbers that we may need to use while simplifying expressions with the order of operations:
| Property | Symbols |
|---|---|
| \text{Commutative property of addition} | a+b=b+a |
| \text{Commutative property of multiplication} | a \cdot b=b \cdot a |
| \text{Associative property of addition} | a+\left(b+c\right) =\left(a+b\right)+c |
| \text{Associative property of multiplication} | a\cdot(b\cdot c)=(a\cdot b)\cdot c |
| \text{Additive identity} | a+0=a \text{ and } 0+a=a |
| \text{Multiplicative identity} | a\cdot 1=a \text{ and }1 \cdot a =a |
| \text{Additive inverse} | a+\left(-a\right)=0 \text{ and } \left(-a\right)+a=0 |
| \text{Multiplicative inverse} | {a \cdot \dfrac{1}{a} =1 \text{ and } \dfrac{1}{a} \cdot a=1} \text{; where }a \neq 0 |
| \text{Multiplicative property of zero} | a \cdot 0 =0 \text{ and }0 \cdot a =0 |
Examples
Example 1
Calculate 86+\dfrac{3}{10}\cdot \left(-2\right)
Example 2
Calculate \sqrt{25-9} + (0.75 \cdot 2^3) - 1.2 \div 0.4
Example 3
Calculate \dfrac{4\cdot\left(\dfrac{1}{16}\cdot2\right)}{2 \cdot4}
Example 4
Calculate 2.3 + |2 - 6.5| - (1.8 + 0.5)
When evaluating multiple operations with rational numbers:
Evaluate operations inside grouping symbols.
Grouping symbols may include parentheses (\text{ }), brackets [ \text{ }], absolute value bars | \text{ }|, square root symbols \sqrt{\text{ }}, and fraction bars \frac{⬚}{⬚}.
Evaluate any exponents.
Evaluate any multiplication or division, reading from left to right.
Evaluate any addition or subtraction, reading from left to right.
Consider and check the sign of the final answer, especially if negative values were involved in the calculation.
Here are some of the properties of real numbers that we may need to use while simplifying expressions with the order of operations:
| Property | Symbols |
|---|---|
| \text{Commutative property of addition} | a+b=b+a |
| \text{Commutative property of multiplication} | a \cdot b=b \cdot a |
| \text{Associative property of addition} | a+\left(b+c\right) =\left(a+b\right)+c |
| \text{Associative property of multiplication} | a\cdot(b\cdot c)=(a\cdot b)\cdot c |
| \text{Additive identity} | a+0=a \text{ and } 0+a=a |
| \text{Multiplicative identity} | a\cdot 1=a \text{ and }1 \cdot a =a |
| \text{Additive inverse} | a+\left(-a\right)=0 \text{ and } \left(-a\right)+a=0 |
| \text{Multiplicative inverse} | {a \cdot \dfrac{1}{a} =1 \text{ and } \dfrac{1}{a} \cdot a=1} \text{; where }a \neq 0 |
| \text{Multiplicative property of zero} | a \cdot 0 =0 \text{ and }0 \cdot a =0 |