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Apply the order of operations with rational numbers

Numerical expressions with rational numbers—including fractions, decimals, and percent representations—are interpreted through grouping symbols, exponents with whole-number powers, multiplication and division, and addition and subtraction, in that hierarchy and from left to right within the same level. The understanding includes preserving the value of negative quantities and equivalent rational forms, recognizing that grouping changes an expression’s meaning, and evaluating nested expressions accurately; symbolic variables, variable exponents, and more advanced exponent rules are outside this scope.

Detailed Explanation: 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:

34−(2(0.5−25%)2÷(−12))\frac{3}{4}-\left(2(0.5-25\%)^2\div\left(-\frac12\right)\right)

Step 1: Rewrite equivalent rational forms.

0.5=12and25%=25100=140.5=\frac12 \qquad\text{and}\qquad 25\%=\frac{25}{100}=\frac14

So the expression becomes

34−(2(12−14)2÷(−12))\frac{3}{4}-\left(2\left(\frac12-\frac14\right)^2\div\left(-\frac12\right)\right)

Step 2: Evaluate the innermost grouping.

12−14=14\frac12-\frac14=\frac14

Now we have

34−(2(14)2÷(−12))\frac{3}{4}-\left(2\left(\frac14\right)^2\div\left(-\frac12\right)\right)

Step 3: Evaluate the exponent.

(14)2=116\left(\frac14\right)^2=\frac{1}{16}

Now we have

34−(2⋅116÷(−12))\frac{3}{4}-\left(2\cdot\frac{1}{16}\div\left(-\frac12\right)\right)

Step 4: Multiply and divide from left to right.

2⋅116=182\cdot\frac{1}{16}=\frac18

Dividing by a negative fraction gives a negative result:

18÷(−12)=18⋅(−2)=−14\frac18\div\left(-\frac12\right) =\frac18\cdot\left(-2\right) =-\frac14

So the expression is now

34−(−14)\frac34-\left(-\frac14\right)

Step 5: Subtract the negative.

34−(−14)=34+14=1\frac34-\left(-\frac14\right) =\frac34+\frac14 =1

Therefore,

1\boxed{1}

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

Learn by doing: Apply the order of operations with rational numbers

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