Number Sense and Operations: Rational Numbers, Integers, and Exponents
What this domain tests
The Number Sense and Operations domain of Grade 7 FAST checks whether you can compute fluently with positive and negative rational numbers, interpret absolute value, and apply exponent rules to numerical expressions. Items often combine two or three skills in one expression, so the order of operations matters.
Core rules
- A rational number can be written as a fraction a/b with b ≠ 0. Its decimal form either terminates or repeats: −3/8 = −0.375, and 2/11 = 0.181818…
- To add or subtract integers with different signs, subtract the smaller absolute value from the larger and keep the sign of the number with the larger absolute value. Subtracting a number is the same as adding its opposite: 5 − (−3) = 5 + 3 = 8.
- A product or quotient is positive when the signs are the same and negative when the signs differ. To divide by a fraction, multiply by its reciprocal.
- Absolute value is distance from zero: |−7| = 7 and |7| = 7. It is never negative, but a minus sign outside the bars stays outside: −|−4| = −4.
- Exponent laws for a ≠ 0: a^m · a^n = a^(m+n), a^m ÷ a^n = a^(m−n), (a^m)^n = a^(mn), and a^0 = 1.
- Order of operations: grouping symbols, exponents, multiplication and division left to right, then addition and subtraction left to right.
Worked example
Evaluate −3/4 ÷ 3/8 + (−2)^3 × 0.5. First rewrite division as multiplication by the reciprocal: −3/4 × 8/3 = −24/12 = −2. Next evaluate the exponent: (−2)^3 = −8, so −8 × 0.5 = −4. Finally add: −2 + (−4) = −6. A common wrong answer is −2 + 4 = 2, which comes from treating (−2)^3 as positive.
Traps
- Applying the exponent only to the numeral and losing the negative sign: (−2)^2 = 4, but −2^2 = −4.
- Subtracting a negative and moving the wrong direction on the number line.
- Forgetting to flip the second fraction when dividing fractions.
- Adding exponents when bases are multiplied but not when bases are added: 2^3 · 2^2 = 2^5, not 4^5.