Secondary 4 Arithmetic Worksheet

All 11 questions

Each question targets one named concept from the sheet. Read them here, or print the PDF — it has working space under each one.

1. Q1Absolute Value Notation –

   Evaluate or solve, showing your steps.

   1. $|-3|·|4-9|$
   2. $|9-2|-|2-9|$
   3. Solve $|2x-5|=11$.
2. Q2Arithmetic (topic overview)

   Each expression below uses a different piece of notation from this topic. Name the notation, then give the value (or the simplest exact form).

   1. $5!$
   2. ${\log }_{3}81$
   3. $|-2.5|$
   4. ${10}^{-2}$
   5. ${\displaystyle \frac{2}{\sqrt{2}}}$
3. Q3Factorial Notation

   Evaluate without a calculator, simplifying before multiplying.

   1. ${\displaystyle \frac{8!}{6!}}$
   2. ${\displaystyle \frac{10!}{3!7!}}$
4. Q4Logarithms

   Evaluate or solve.

   1. ${\log }_{2}64$
   2. ${\log }_5{\displaystyle \frac{1}{25}}$
   3. Solve ${\log }_{3}x=4$.
5. Q5Operations

   Nadia and Owen both evaluate $12-4\times (5-8{)}^2÷6$. Nadia gets $6$; Owen gets $12$. Decide who is correct, show the correct evaluation step by step, and state precisely which rule the other student broke.

6. Q6Rationalizing Fractions

   Rationalize the denominator and simplify.

   1. ${\displaystyle \frac{6}{\sqrt{3}}}$
   2. ${\displaystyle \frac{5}{\sqrt{10}}}$
7. Q7Representing Number Sets –

   Write each set in interval notation.

   1. all real numbers greater than or equal to $-3$ and strictly less than $7$
   2. $\{x\in ℝ\mid x>2\}$
   3. $[-1,5]\cap ]3,+\infty [$
8. Q8The Common Denominator –

   Add or subtract, giving the result in lowest terms.

   1. ${\displaystyle \frac{3}{4}}+{\displaystyle \frac{5}{6}}-{\displaystyle \frac{7}{12}}$
   2. ${\displaystyle \frac{2}{3x}}+{\displaystyle \frac{5}{6x}}$, and state the restriction on $x$.
9. Q9The Laws of Exponents

   Simplify, leaving no negative exponents.

   1. ${\displaystyle \frac{{\left(2a^3{b}^{-2}\right)}^2·a^{-1}}{4a^2{b}^{-5}}}$
   2. Evaluate ${\left({\displaystyle \frac{27}{8}}\right)}^{-2/3}$.
10. Q10The Laws of Logarithms

    Use the laws of logarithms.

    1. Write $2{\log }_{3}6-{\log }_{3}4$ as a single logarithm, then evaluate it.
    2. Given $\log 2\approx 0.301$ and $\log 3\approx 0.477$, find $\log 12$ without a calculator.
11. Q11Writing Numbers

    For each number, state whether it is rational or irrational and justify in one short sentence; then write $0.\overline{36}$ as a fraction in lowest terms. $$0.\overline{36}\sqrt{49}\frac{\pi }{2}2.5\times {10}^{-3}$$

The 10 challenge problems for this topic are a separate, paid sheet and are not reproduced here.

How to do every concept on this sheet

This is the part worksheet sites usually leave out. Below is the actual reasoning behind each group of questions — not a full solution set, the decisions that get you to one. Read it before you start, or after you get stuck.

Read the notation before you compute anything

Q2 puts five different pieces of notation side by side and asks you to name each one before evaluating it, and that is not a warm-up — it is the point of the whole topic. Almost every error on this sheet is a misread symbol rather than bad arithmetic: an exclamation mark treated as emphasis, a negative exponent treated as a negative number, a logarithm treated as multiplication.

Factorials: cancel first, multiply never (Q3)

The instruction "simplify before multiplying" is the entire technique. The identity that does all the work is n! = n · (n − 1)!, applied as many times as you need to expose the smaller factorial and cancel it:

A quotient of factorials always collapses to a short product of consecutive integers, and if yours has not, you expanded too far. Working out 10! in full and dividing is not just slow — it overflows a calculator display and loses precision, and questions in this unit are written to be done without one.

Logarithms are exponents wearing a different hat (Q4)

The definition is a translation, and it is worth writing out every single time until you stop needing to:

That is the whole of Q4. "What is log₂ 64?" becomes "2 to the what is 64?", which you can answer by counting. "Solve log₃ x = 4" becomes "x = 3⁴", which is not really an equation to solve at all. And log₅(1⁄25) comes out negative because reaching a number smaller than 1 needs a negative exponent — a logarithm being negative is normal and says nothing alarming.

The laws of logarithms: combine, then evaluate (Q10)

Three laws, all of them the exponent laws seen from the other side. A product inside becomes a sum outside, a quotient becomes a difference, and an exponent inside comes out to the front as a coefficient. Q10(a) runs them backwards: pull the coefficient back inside as a power, combine into a single logarithm, and only then evaluate — because log₃ 9 is something you can answer and 2 log₃ 6 is not.

Q10(b) is the same idea used as a computational trick. Given only log 2 and log 3, you can reach any number built from 2s and 3s by factoring it first: 12 is 2² · 3, so log 12 = 2 log 2 + log 3. Factoring the argument before applying a law is the move that makes these questions possible without a calculator.

Exponent laws, and what a fractional exponent means (Q9)

Q9(a) is bookkeeping: apply the power to every factor inside the bracket including the numerical coefficient, add exponents when multiplying like bases, subtract when dividing, and finish by clearing negative exponents. "Leaving no negative exponents" is part of the instruction, so a technically correct answer with a b⁻¹ in it is not finished.

Q9(b) is the more interesting half. A fractional exponent packs a root and a power into one symbol: a^(m⁄n) is the nth root of a, raised to m. Add a minus sign and you also flip. The order you do the three operations in is free — the result is the same — but one order keeps the numbers small:

Order of operations, and the two "left to right" pairs (Q5)

Q5 hands you two students with two answers and asks you to referee, which is harder than doing the computation yourself because you have to name the broken rule. The order is brackets, then exponents, then multiplication and division, then addition and subtraction — with one clause people forget: multiplication and division sit at the same level and are done left to right, and so do addition and subtraction. Division does not wait for every multiplication to finish.

The failure in Q5 is a different one, and it is the classic: doing a subtraction that appears first on the page before the multiplication and division that outrank it. Reading order is not priority order. When you write up the answer, name the rule that was broken rather than just showing the correct chain — that is what the question is marked on.

Rationalizing a denominator (Q6)

Multiply the fraction by the radical over itself. That is multiplying by 1, so the value never changes; only the way it is written does, and √a · √a = a clears the denominator.

Two habits: simplify the radical before you rationalize when it hides a perfect square factor — √48 is 4√3, and rationalizing before spotting that makes the numbers four times bigger than they need to be — and always reduce the fraction at the end, because the numerator and the cleared denominator usually share a factor, as they do above. When the denominator is a sum containing a radical, the same idea works with the conjugate, since (p − √q)(p + √q) = p² − q has no radical left in it.

Absolute value: a distance, and an equation with two answers (Q1)

|x| is how far x sits from zero, which is why it is never negative and why the order of a subtraction inside it is irrelevant. Treat the bars as brackets: finish the arithmetic inside, then take the size.

Solving is where the structure matters. |u| = 11 says "u is 11 away from zero", and two numbers satisfy that, so the equation splits into u = 11 and u = −11, each solved separately. Two solutions is the normal outcome, one is possible when the right-hand side is zero, and none at all when the right-hand side is negative — which is worth checking before you split anything, because a distance is never negative.

Intervals, common denominators and the number sets (Q7, Q8, Q11)

In Quebec notation a bracket turns outward at an endpoint that is excluded, so [−3, 7[ includes −3 and stops just short of 7, and infinity always gets a reversed bracket because it is never reached. An intersection, as in Q7(c), keeps only the numbers satisfying both conditions at once — sketch the two intervals on the same number line and read off the overlap rather than trying to reason it out symbolically.

Q8 adds a variable to a familiar operation. The method is unchanged — build the smallest common denominator, rewrite each fraction, combine the numerators — but the answer is incomplete without the restriction. A denominator of 3x or 6x requires x ≠ 0, and that condition comes from the denominators you started with, not from the simplified result.

Q11 asks which numbers are rational. A number is rational exactly when it can be written as a ratio of two integers, and three signals cover almost every case you will meet: a terminating decimal is rational, a repeating decimal is rational, and a root that comes out whole is rational. Everything else needs an argument rather than an impression — and a non-zero rational multiple of an irrational number stays irrational, which is the sentence Q11 is fishing for. To convert a repeating decimal, put the repeating block over as many 9s as it has digits, then reduce.

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Getting the most out of it

Do the notation question twice

Answer the overview question cold before anything else, and mark honestly which of the five symbols you could not name. Those are your revision list. Answer it again after the rest of the sheet and the difference is a fair measure of what actually stuck.

Put the calculator away

Every question here is designed to come out exactly: the factorials cancel, the logarithms land on whole numbers, the roots are perfect. Reaching for a calculator hides whether you understood the notation, and the exam version of these questions is written the same way.

Write the exponential form under every logarithm

Under log₂ 64 = ?, write 2^? = 64. It is one extra line and it converts a piece of unfamiliar notation into a question you have been able to answer since Secondary 1.

Common questions