Grade 7A–8A · Math Exam Prep

Tricky Math
Questions

Most-missed exam questions on Rational Numbers, Real Numbers, and Exponents — with instant feedback.

7A · 2.5 Rational Numbers 7A · 2.6 Real Numbers 8A · 1.1 Positive Exponents 8A · 1.2 Zero & Negative Exponents 8A · 1.4 Comparing Exponents
0 / 20 answered
Score: 0
UNIT 7A · 2.5

Rational Numbers

⚡ Quick Memory Points

FRACTION → DECIMAL Divide numerator ÷ denominator. Terminating = ends. Repeating = has a pattern (bar notation).
RATIONAL Any number writable as $\frac{p}{q}$ where $p, q$ are integers and $q \neq 0$. Includes ALL integers, fractions, terminating & repeating decimals.
DENSITY Between ANY two rationals, there are INFINITELY many more. Always find one by averaging: $\frac{a+b}{2}$.
ORDERING TRICK Convert everything to decimals first — easiest way to compare mixed forms.
Q 01 TRICKY
Which of the following is NOT a rational number? ⚠ Most students pick the wrong one because they confuse repeating decimals with irrationals.
📖 Explanation Answer: C — $\sqrt{5}$

$0.\overline{3} = \frac{1}{3}$ ✓ rational. $-\frac{7}{2}$ ✓ rational. $0.125 = \frac{1}{8}$ ✓ rational.
$\sqrt{5} \approx 2.2360679…$ — this decimal never terminates and never repeats, so it cannot be written as $\frac{p}{q}$. It is irrational.
Q 02 HARD
Write $0.\overline{27}$ as a fraction in lowest terms. Classic exam trap — many students forget to simplify.
📖 Explanation Answer: B — $\dfrac{3}{11}$

Let $x = 0.\overline{27}$. Then $100x = 27.\overline{27}$.
Subtract: $99x = 27 \Rightarrow x = \dfrac{27}{99}$.
Simplify: $\gcd(27,99) = 9$, so $\dfrac{27}{99} = \dfrac{3}{11}$. ✓
Trap: Option C is correct but not fully simplified — always check lowest terms!
Q 03 TRICKY
How many rational numbers exist between $\dfrac{1}{3}$ and $\dfrac{1}{2}$?
📖 Explanation Answer: D — Infinitely many

This is the Density Property of rational numbers. Between any two distinct rationals, you can always find another by averaging: $\dfrac{\frac{1}{3}+\frac{1}{2}}{2} = \dfrac{5}{12}$. Repeat forever — there's no limit.
Q 04 HARD
Order from least to greatest: $\quad -\dfrac{3}{4},\quad -0.8,\quad -\dfrac{5}{6},\quad -0.7\overline{5}$
📖 Explanation Answer: B

Convert to decimals: $-\frac{3}{4} = -0.75$, $-0.8$, $-\frac{5}{6} \approx -0.8\overline{3}$, $-0.7\overline{5} = -0.7555…$
On a number line, more negative = smaller: $-0.8\overline{3} < -0.8 < -0.7\overline{5} < -0.75$
So: $-\frac{5}{6},\ -0.8,\ -0.7\overline{5},\ -\frac{3}{4}$ ✓
Unit 7A · 2.6 — Real Numbers & Calculators

⚡ Quick Memory Points

IRRATIONAL Non-terminating AND non-repeating decimal. $\sqrt{2}, \sqrt{3}, \pi, e$ — cannot be expressed as $\frac{p}{q}$.
REAL = R + I Real Numbers = Rationals ∪ Irrationals. Every number on the number line is real.
PERFECT SQUARE $\sqrt{n}$ is rational ONLY if $n$ is a perfect square (1,4,9,16,25…). Otherwise irrational.
CALCULATOR TRAP Calculator shows a rounded value — NOT the exact irrational. e.g., $\sqrt{2} \neq 1.414$, it just ≈ that.
Q 05 TRICKY
A student's calculator shows $\sqrt{2} = 1.41421356$. The student writes: "Therefore $\sqrt{2}$ is rational." What is wrong?
📖 Explanation Answer: B

Calculators have limited display digits. $\sqrt{2} = 1.41421356237…$ continues forever without any repeating pattern. A finite display is just an approximation. The number is provably irrational (Pythagoras proved this ~500 BC).
Q 06 HARD
Which set correctly classifies $\sqrt{16}$? Students often assume $\sqrt{\ }$ always means irrational.
📖 Explanation Answer: B

$\sqrt{16} = 4$. Since 16 is a perfect square, its square root is a whole number.
$4$ is an integer → also rational ($= \frac{4}{1}$) → also real. Every integer is rational, every rational is real.
Q 07 HARD
Estimate $\sqrt{50}$ to the nearest tenth without a calculator.
📖 Explanation Answer: C — $7.1$

$7^2 = 49$ and $8^2 = 64$, so $\sqrt{50}$ is between $7$ and $8$, very close to $7$.
Check: $7.1^2 = 50.41$ and $7.0^2 = 49.00$. Since $50$ is much closer to $49$, the answer rounds to $7.1$.
Unit 8A · 1.1 — Positive Exponents & Laws

⚡ Quick Memory Points

PRODUCT RULE $a^m \cdot a^n = a^{m+n}$ — same base: ADD exponents.
QUOTIENT RULE $\dfrac{a^m}{a^n} = a^{m-n}$ — same base dividing: SUBTRACT exponents.
POWER RULE $(a^m)^n = a^{m \times n}$ — power of a power: MULTIPLY exponents.
POWER of PRODUCT $(ab)^n = a^n b^n$ — distribute the exponent to EACH factor.
MNEMONIC: "Add · Sub · Mult" Product→Add, Quotient→Subtract, Power→Multiply. Say it 3 times!
Q 08 TRICKY
Simplify: $\quad (2x^3)^4$ ⚠ Most common error: forgetting to raise the coefficient to the power too.
📖 Explanation Answer: C — $16x^{12}$

$(2x^3)^4 = 2^4 \cdot (x^3)^4 = 16 \cdot x^{3 \times 4} = 16x^{12}$
The power distributes to BOTH the 2 AND $x^3$. $2^4 = 16$, not $2 \times 4 = 8$.
Q 09 HARD
Simplify: $\quad \dfrac{x^5 \cdot x^3}{x^4}$
📖 Explanation Answer: B — $x^4$

Numerator: $x^5 \cdot x^3 = x^{5+3} = x^8$ (Product Rule: add)
Then: $\dfrac{x^8}{x^4} = x^{8-4} = x^4$ (Quotient Rule: subtract)
Work step by step — don't skip steps under exam pressure!
Q 10 HARD
Simplify: $\quad \left(\dfrac{3a^2}{b^3}\right)^3$
📖 Explanation Answer: C — $\dfrac{27a^6}{b^9}$

Apply power to numerator AND denominator:
$3^3 = 27$, $(a^2)^3 = a^6$, $(b^3)^3 = b^9$
Result: $\dfrac{27a^6}{b^9}$ ✓
Unit 8A · 1.2 — Zero & Negative Exponents

⚡ Quick Memory Points

ZERO POWER $a^0 = 1$ for any $a \neq 0$. Even $(-999)^0 = 1$. The only exception: $0^0$ is undefined.
NEGATIVE = FLIP $a^{-n} = \dfrac{1}{a^n}$ — negative exponent means take the RECIPROCAL. Move it to the other side of the fraction.
FLIP TRICK $\dfrac{a^{-m}}{b^{-n}} = \dfrac{b^n}{a^m}$ — negative exponents "cross over" the fraction bar and become positive.
NEVER NEGATIVE VALUE $a^{-n}$ is NOT negative — it's $\frac{1}{a^n}$, which is a positive fraction (if $a > 0$). The minus is in the EXPONENT, not the value!
Q 11 TRICKY
Evaluate: $\quad 5^0 + 5^{-1}$ ⚠ Students often compute $5^0 = 0$ or $5^{-1} = -5$. Both wrong!
📖 Explanation Answer: C — $\dfrac{6}{5}$

$5^0 = 1$ (anything to the zero power is 1)
$5^{-1} = \dfrac{1}{5}$ (negative exponent = reciprocal)
Sum: $1 + \dfrac{1}{5} = \dfrac{5}{5} + \dfrac{1}{5} = \dfrac{6}{5}$ ✓
Q 12 HARD
Simplify and write with positive exponents only: $\quad \dfrac{4x^{-3}}{y^{-2}}$
📖 Explanation Answer: B — $\dfrac{4y^2}{x^3}$

$x^{-3}$ in the numerator moves to the denominator as $x^3$.
$y^{-2}$ in the denominator moves to the numerator as $y^2$.
$\dfrac{4x^{-3}}{y^{-2}} = \dfrac{4 \cdot y^2}{x^3} = \dfrac{4y^2}{x^3}$ ✓
Rule: Negative exponents CROSS the fraction bar and become positive.
Q 13 HARD
Which expression equals $\left(\dfrac{2}{3}\right)^{-2}$?
📖 Explanation Answer: C — $\dfrac{9}{4}$

$\left(\dfrac{2}{3}\right)^{-2} = \left(\dfrac{3}{2}\right)^{2} = \dfrac{9}{4}$
A negative exponent on a fraction: flip the fraction first, then apply the positive exponent. The result is always positive (for positive bases).
Unit 8A · 1.4 — Comparing Exponents

⚡ Quick Memory Points

SAME BASE $a^m$ vs $a^n$ with $a > 1$: bigger exponent = bigger value. With $0 < a < 1$: bigger exponent = SMALLER value (fractions shrink).
COMMON BASE Rewrite everything in the same base before comparing. e.g., $4^3 = (2^2)^3 = 2^6$.
SCI NOTATION For very large/small numbers, compare the powers of 10 first. Bigger power of 10 = bigger number (if coefficient ≥ 1).
FRACTION BASE FLIP $\left(\frac{1}{2}\right)^{10} < \left(\frac{1}{2}\right)^2$ — when base is a proper fraction, bigger exponent makes it SMALLER!
Q 14 TRICKY
Which is greater: $\left(\dfrac{1}{3}\right)^4$ or $\left(\dfrac{1}{3}\right)^2$ ? ⚠ Most students incorrectly choose $\left(\frac{1}{3}\right)^4$ because $4 > 2$.
📖 Explanation Answer: B

$\left(\frac{1}{3}\right)^2 = \frac{1}{9} \approx 0.111$
$\left(\frac{1}{3}\right)^4 = \frac{1}{81} \approx 0.012$
When the base is a proper fraction ($0 < a < 1$), higher powers make it smaller. Think of it as multiplying by $\frac{1}{3}$ repeatedly — each multiplication makes it tinier.
Q 15 HARD
Compare: $\quad 2^{10}$ vs $10^3$
📖 Explanation Answer: A

$2^{10} = 2 \times 2 \times … \times 2 = 1024$
$10^3 = 1000$
$1024 > 1000$, so $2^{10} > 10^3$. Don't assume the bigger base always wins — the exponent matters too!
Q 16 HARD
Which is largest? $\quad 4^3,\quad 2^7,\quad 8^2$
📖 Explanation Answer: B — $2^7 = 128$

Convert to base 2: $4^3 = (2^2)^3 = 2^6 = 64$, $\quad 8^2 = (2^3)^2 = 2^6 = 64$, $\quad 2^7 = 128$
Useful trick: rewrite all in the same base (here: base 2) to compare directly. $2^7 > 2^6$, so $2^7$ wins.
Q 17 HARD
Order from least to greatest: $\quad 3^{-2},\quad 2^{-3},\quad 4^{-1}$
📖 Explanation Answer: B

Convert to fractions: $3^{-2} = \frac{1}{9} \approx 0.111$, $\quad 2^{-3} = \frac{1}{8} = 0.125$, $\quad 4^{-1} = \frac{1}{4} = 0.25$
Order: $\frac{1}{9} < \frac{1}{8} < \frac{1}{4}$, i.e., $2^{-3} < 3^{-2} < 4^{-1}$ ✓
Remember: For unit fractions $\frac{1}{n}$, bigger denominator = smaller value.
Q 18 TRICKY
Simplify: $\quad \dfrac{2^3 \cdot 2^{-5}}{2^{-1}}$
📖 Explanation Answer: B — $\dfrac{1}{2}$

Numerator: $2^3 \cdot 2^{-5} = 2^{3+(-5)} = 2^{-2}$
Then: $\dfrac{2^{-2}}{2^{-1}} = 2^{-2-(-1)} = 2^{-2+1} = 2^{-1} = \dfrac{1}{2}$ ✓
Key: When dividing, subtract exponents — and subtracting a negative = adding!
Q 19 HARD
A bacterium doubles every hour. Starting with $1$ bacterium, which expression gives the count after $n$ hours? Real-world exponent application — common in exams.
📖 Explanation Answer: C — $2^n$

Hour 0: $1 = 2^0$. Hour 1: $2 = 2^1$. Hour 2: $4 = 2^2$. Hour 3: $8 = 2^3$.
This is exponential growth: the count multiplies by 2 each hour, giving $2^n$.
Linear ($2n$) would give 2, 4, 6, 8 — which is wrong (additive not multiplicative).
Q 20 HARD
Which is equivalent to $\dfrac{1}{x^{-3} \cdot x^5}$? ⚠ Multi-step — students rush and make sign errors.
📖 Explanation Answer: B — $\dfrac{1}{x^2}$

Denominator: $x^{-3} \cdot x^5 = x^{-3+5} = x^2$
So: $\dfrac{1}{x^2} = x^{-2}$ ✓
Step 1: Simplify the denominator using product rule. Step 2: Then flip it. Don't try to skip steps!

FINAL SCORE

—

0 CORRECT
0 WRONG
0% ACCURACY