How do you tell whether a number is rational or irrational?

A rational number can be written as a fraction of two integers, and its decimal terminates or repeats. An irrational number cannot, and its decimal runs forever without repeating. Square roots of non-perfect-squares (like the square root of 7), pi, and cube roots of non-perfect-cubes are irrational, but check first: the square root of 16 is 4, which is rational. A rational plus an irrational is always irrational, and a nonzero rational times an irrational is irrational.

What is the difference between the product rule and multiplying coefficients?

When you multiply powers of the same base you add the exponents, but you multiply the coefficients normally. So 3 times x to the fourth, times 2 times x to the fifth, gives 6 x to the ninth: multiply 3 and 2 to get 6, and add 4 and 5 to get 9. A frequent MCAS error is multiplying the exponents to get x to the twentieth, which is wrong.

What does a negative exponent mean?

A negative exponent signals a reciprocal, not a negative number. Two to the negative third is one over two cubed, which is one eighth, a positive value. For a fraction raised to a negative power, invert the fraction and make the exponent positive: two thirds to the negative two equals three halves squared, which is nine fourths. A fully simplified answer has no negative exponents left.

How do you simplify a square root to simplest form?

Factor out the largest perfect-square factor of the radicand and take its root outside. The square root of 72 is the square root of 36 times 2, which is 6 times the square root of 2. If you do not see the largest factor at once, peel off smaller squares and repeat until nothing inside is a perfect square. An answer like 2 times the square root of 18 is not simplest because 18 still contains the perfect square 9.

Why does a 25 percent markup then a 25 percent discount not return the original price?

Because the discount acts on the higher price, not the original. An 80 dollar item marked up 25 percent becomes 100 dollars; a 25 percent discount from 100 is 75 dollars, not 80. Each percent change multiplies the running base, and the two bases differ, so equal-percent up and down give a net decrease. The multipliers are 1.25 and 0.75, and their product is 0.9375, a 6.25 percent net drop.

How do you decide which way to round in a context problem?

Match the rounding to what the question asks. How many objects fit or can be made rounds down, because a partial one does not count. How many are needed to cover a demand rounds up, because a partial one still requires a whole. The same decimal, like 5.56, can mean 5 full tiles that fit or 6 buses that are needed, depending on the wording.

MassachusettsMaths

Grade 10 Math MCAS: a complete guide to the Number and Quantity category

A deep-dive Grade 10 Math MCAS guide to the Number and Quantity category. Covers classifying rational and irrational numbers, the laws of exponents with zero and negative powers, simplifying radicals and rational exponents, unit analysis and precision, and proportional reasoning with percent change, plus the no-calculator skills the MCAS rewards.

Generated by Claude Opus 4.814 min readN-RN, N-QUpdated 2026-06-13

Reviewed by: AI editorial process; not yet individually human-reviewed

Jump to a section

What this category demands
The real number system
The laws of exponents
Radicals and rational exponents
Units, quantities, and precision
Proportional reasoning and percent change
How this category is examined
Check your knowledge

What this category demands

The Number and Quantity category is the numerical foundation of the Grade 10 MCAS, drawn from the N-RN and N-Q standards of the 2017 Massachusetts Curriculum Framework for Mathematics. It is not the largest category by item count, but its skills (exponents, radicals, unit work, and percent) reappear everywhere else: in the quadratic formula, the Pythagorean theorem, exponential models, and statistics. Much of it must be done in the no-calculator session, so fluency by hand is the goal. This guide ties together the dot-point pages, each with its own practice: the real number system and rational versus irrational, properties of exponents, radicals and rational exponents, units, quantities, and precision, and ratios, rates, and proportional reasoning.

The real number system

Every real number is either rational (a ratio of integers, with a terminating or repeating decimal) or irrational (neither terminating nor repeating). The MCAS tests both classification and the closure rules:

\text{rational} + \text{rational} = \text{rational}, \qquad \text{rational} + \text{irrational} = \text{irrational}, \qquad \text{nonzero rational} \times \text{irrational} = \text{irrational}.

Two irrationals are unpredictable: $\sqrt{2} \cdot \sqrt{2} = 2$ is rational, while $\sqrt{2} + \sqrt{2} = 2\sqrt{2}$ is irrational. Always check a radical before calling it irrational, since $\sqrt{36} = 6$ .

The laws of exponents

The exponent laws follow from repeated multiplication: $x^m \cdot x^n = x^{m+n}$ , $\dfrac{x^m}{x^n} = x^{m-n}$ , $(x^m)^n = x^{mn}$ , $x^0 = 1$ , and $x^{-n} = \dfrac{1}{x^n}$ . The two ideas the MCAS most rewards are that a negative exponent means reciprocal (so $4^{-2} = \frac{1}{16}$ , positive) and that a fully simplified expression has no negative exponents. Watch the difference between adding exponents and multiplying coefficients: $3x^4 \cdot 2x^5 = 6x^9$ .

Radicals and rational exponents

Simplify a radical by pulling out the largest perfect-square (or cube) factor: $\sqrt{72} = 6\sqrt{2}$ . Combine radicals only when the radicands match: $\sqrt{8} + \sqrt{18} = 2\sqrt{2} + 3\sqrt{2} = 5\sqrt{2}$ . Rational exponents are the same idea in different notation, with the denominator as the root index: $a^{m/n} = \sqrt[n]{a^m}$ , so $27^{2/3} = \left(\sqrt[3]{27}\right)^2 = 9$ . Evaluate root-first to keep numbers small without a calculator.

Units, quantities, and precision

Convert with unit fractions arranged so unwanted units cancel: $60 \text{ mph} \times \frac{5280 \text{ ft}}{1 \text{ mi}} \times \frac{1 \text{ hr}}{3600 \text{ s}} = 88 \text{ ft/s}$ . Carry units through every line; if the leftover unit is not the target, the setup is inverted. Report answers with context-appropriate precision: money to the cent, counts to whole numbers, and no more decimal places than the data supports.

Proportional reasoning and percent change

Solve a proportion by cross-multiplying $\frac{a}{b} = \frac{c}{d}$ into $ad = bc$ , keeping matching units in matching positions. Treat percent change as a multiplier: a $p\%$ increase is $\times (1 + \frac{p}{100})$ , a decrease is $\times (1 - \frac{p}{100})$ , and the percent change between two values is $\frac{\text{new} - \text{old}}{\text{old}} \times 100$ on the original base. Successive percents act on changing bases, so a 25% markup and 25% discount net to $1.25 \times 0.75 = 0.9375$ , a 6.25% decrease.

How this category is examined

Selected-response (1 point, often no calculator). Classify a number, simplify an exponent or radical expression, pick the better unit rate, or compute a percent change. Multiple-select may ask for all equivalent forms.
Short-answer (1 point). A single conversion, a single percent step, or a simplified radical, scored by exact match.
Constructed-response (within multi-step items). Unit work and percent change usually appear as the setup inside a larger modeling problem; getting the units right is what lets the rest of the solution earn credit.

Check your knowledge

Work these as you would for credit, and do the starred ones without a calculator.

Is $\sqrt{3} + \sqrt{12}$ rational or irrational? Justify. (2 points)
Simplify $\dfrac{15x^6 y^3}{5x^2 y^7}$ with no negative exponents. (1 point)*
Evaluate $\left(\dfrac{4}{9}\right)^{-1/2}$ . (2 points)*
Simplify $\sqrt{200} - \sqrt{50}$ . (2 points)*
Convert 45 miles per hour to feet per second. (2 points)
A 10-pound bag costs \ $12.50 and a 4-pound bag costs \$ 5.20. Which is the better unit price? (2 points)
A price rises from \ $60 to \$ 75. What is the percent increase? (1 point)*
A $120 item is discounted 20%, then taxed 5% on the sale price. What is the final cost? (2 points)

Solutions

Step 1: Classify the sum of two square roots (Q1)

Simplify $\sqrt{12}$ first by pulling out its perfect-square factor: $\sqrt{12} = \sqrt{4 \cdot 3} = 2\sqrt{3}$ . The sum becomes $\sqrt{3} + 2\sqrt{3} = 3\sqrt{3}$ . The number $3$ is a nonzero rational, and $\sqrt{3}$ is irrational, so their product is irrational by the closure rule.

Step 2: Simplify a quotient of powers with no negative exponents (Q2)

Divide the coefficients and apply the quotient rule for each variable separately. The coefficients give $\frac{15}{5} = 3$ . For $x$ , the exponents subtract: $x^{6-2} = x^4$ . For $y$ , the exponents subtract: $y^{3-7} = y^{-4}$ . A negative exponent means a reciprocal, so move $y^{-4}$ to the denominator:

\frac{15x^6 y^3}{5x^2 y^7} = \frac{3x^4}{y^4}.

Step 3: Evaluate a negative fractional exponent (Q3)

A negative exponent inverts the base, and the denominator of the exponent is the root index. Invert the fraction first, then take the square root:

\left(\frac{4}{9}\right)^{-1/2} = \left(\frac{9}{4}\right)^{1/2} = \frac{3}{2}.

Step 4: Subtract simplified radicals (Q4)

Rewrite each radical by extracting its largest perfect-square factor. For $\sqrt{200}$ : $\sqrt{200} = \sqrt{100 \cdot 2} = 10\sqrt{2}$ . For $\sqrt{50}$ : $\sqrt{50} = \sqrt{25 \cdot 2} = 5\sqrt{2}$ . The radicands now match, so subtract the coefficients:

\sqrt{200} - \sqrt{50} = 10\sqrt{2} - 5\sqrt{2} = 5\sqrt{2}.

Step 5: Convert miles per hour to feet per second (Q5)

Multiply by unit fractions so that miles and hours cancel and feet and seconds remain. There are $5280$ feet in a mile and $3600$ seconds in an hour:

45 \times \frac{5280}{3600} = 45 \times \frac{22}{15} = 66 \text{ feet per second.}

Step 6: Compare unit prices (Q6)

Divide each total price by the corresponding weight to find the price per pound. For the 10-pound bag: $\frac{12.50}{10} = 1.25$ dollars per pound. For the 4-pound bag: $\frac{5.20}{4} = 1.30$ dollars per pound. Since $1.25 < 1.30$ , the 10-pound bag is the better unit price.

Step 7: Find the percent increase (Q7)

Use the percent-change formula with the original price as the base. The increase is $75 - 60 = 15$ dollars:

\frac{75 - 60}{60} \times 100 = \frac{15}{60} \times 100 = 25\%.

Step 8: Apply successive percent changes (Q8)

Each percent change is applied as a multiplier to the running total. A 20% discount multiplies by $0.80$ , and a 5% tax then multiplies by $1.05$ :

120 \times 0.80 = 96, \qquad 96 \times 1.05 = 100.80.

Final answer: $100.80.

Sources & how we know this

Release of Spring 2025 MCAS Test Items: Grade 10 Mathematics — Massachusetts DESE (2025)
MCAS Grade 10 Mathematics Reference Sheet — Massachusetts DESE (2024)