Skill: Apply the order of operations with rational numbers

Use this order:

1. Grouping symbols: parentheses, brackets, or braces
2. Exponents with whole-number powers
3. Multiplication and division, from left to right
4. Addition and subtraction, from left to right

You can rewrite decimals and percents as fractions when that makes the work easier. Be careful to keep negative signs attached to their numbers.

Evaluate:

$$\frac{3}{4}-\left(2(0.5-25\%)^2\div\left(-\frac12\right)\right)$$

Step 1: Rewrite equivalent rational forms.

$$0.5=\frac12 \qquad\text{and}\qquad 25\%=\frac{25}{100}=\frac14$$

So the expression becomes

$$\frac{3}{4}-\left(2\left(\frac12-\frac14\right)^2\div\left(-\frac12\right)\right)$$

Step 2: Evaluate the innermost grouping.

$$\frac12-\frac14=\frac14$$

Now we have

$$\frac{3}{4}-\left(2\left(\frac14\right)^2\div\left(-\frac12\right)\right)$$

Step 3: Evaluate the exponent.

$$\left(\frac14\right)^2=\frac{1}{16}$$

Now we have

$$\frac{3}{4}-\left(2\cdot\frac{1}{16}\div\left(-\frac12\right)\right)$$

Step 4: Multiply and divide from left to right.

$$2\cdot\frac{1}{16}=\frac18$$

Dividing by a negative fraction gives a negative result:

$$\frac18\div\left(-\frac12\right) =\frac18\cdot\left(-2\right) =-\frac14$$

So the expression is now

$$\frac34-\left(-\frac14\right)$$

Step 5: Subtract the negative.

$$\frac34-\left(-\frac14\right) =\frac34+\frac14 =1$$

Therefore,

$$\boxed{1}$$

Keep the grouping symbols in mind: changing them can change the value of the expression.