Number Sense and Operations
What this domain tests
The Number Sense and Operations domain tests flexible work with rational and irrational numbers, integer and fraction arithmetic, order of operations, exponents and roots, scientific notation, ratios, rates, percents, and elementary number theory. Questions often ask for an equivalent form or for reasoning about whether a result is rational.
Core rules
- Follow order of operations: grouping, exponents and roots, multiplication and division left to right, then addition and subtraction left to right.
- For fractions, a/b ÷ c/d = a/b × d/c; simplify before multiplying when possible.
- Negative exponents mean reciprocals: x^(-n)=1/x^n for x≠0. A negative base with an even exponent is positive; with an odd exponent it is negative.
- A rational number can be written as a fraction of integers; irrational numbers such as √2 and π cannot. Also, √(a²b)=a√b for a≥0.
- Percent change = (new−old)/old ×100%. Percent of a number uses the decimal or fraction equivalent.
Worked example
Simplify (2/3)^(-2) ÷ (4/9). First, (2/3)^(-2) = 1/(2/3)^2 = 1/(4/9)=9/4. Then divide by 4/9: (9/4)÷(4/9)=(9/4)(9/4)=81/16. The result is rational because it is a ratio of integers.
Traps
Do not distribute an exponent over addition: (a+b)^2≠a²+b². Do not cancel terms across addition in a fraction. In percent problems, identify the base: a 20% increase followed by a 20% decrease does not return to the original value. When simplifying radicals, check that the radicand is nonnegative in the real-number context.