Secondary 5 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 –

   At a municipal water plant, a batch of treated water is accepted only if its pH reading $p$ differs from the target value $7.2$ by at most $0.4$.

   1. Write this acceptance condition using absolute value notation.
   2. Give the set of acceptable readings in interval notation.
   3. A batch reads $p=6.75$. Is it accepted? Support your answer with a calculation.
2. Q2Arithmetic

   Each expression below can be written in a simpler exact form using one main arithmetic tool from this topic (laws of exponents, laws of logarithms, rationalizing a denominator, absolute value). For each one, name the tool you would use and give the simplified exact value.

   1. ${\displaystyle \frac{6}{\sqrt{5}-\sqrt{2}}}$
   2. ${\log }_{3}45-{\log }_{3}5$
   3. ${\left(8^{\frac{2}{3}}\right)}^{-2}$
   4. $|-7|-|3-11|$
3. Q3Logarithms

   Evaluate without a calculator, showing the exponential form you used in each case.

   1. ${\log }_2{\displaystyle \frac{1}{32}}$
   2. ${\log }_{9}27$
   3. ${\log }_{0.5}8$
4. Q4Operations

   Evaluate, respecting the priority of operations: $$E=-2^4+(-2{)}^4-\sqrt[3]{-27}\times |-5+2|.$$ Then explain, in one sentence, why $-2^4$ and $(-2{)}^4$ do not have the same value.

5. Q5Rationalizing Fractions

   Write each quotient with a rational denominator, in simplest exact form.

   1. ${\displaystyle \frac{10}{\sqrt{8}}}$
   2. ${\displaystyle \frac{3}{2+\sqrt{7}}}$
6. Q6Representing Number Sets –

   A ferry accepts vehicles whose mass is at least $1.5$ t and less than $8$ t.

   1. Write the set of admissible masses (in tonnes) in interval notation and in set-builder notation.
   2. List, in extension, the elements of $\mathbb{Z}\cap ]-3,4]$.
7. Q7The Laws of Exponents

   Simplify, giving your answers with positive exponents only.

   1. ${\displaystyle \frac{{\left(2a^3{b}^{-2}\right)}^4}{8a^5{b}^{-3}}}$
   2. ${\left({\displaystyle \frac{27}{64}}\right)}^{-\frac{2}{3}}$
8. Q8The Laws of Logarithms

   Write each expression as a single logarithm, then evaluate it exactly.

   1. ${\log }_{2}40+{\log }_{2}6-{\log }_{2}15$
   2. ${\displaystyle \frac{1}{2}}{\log }_{7}81+{\log }_7{\displaystyle \frac{7}{9}}$
9. Q9Using the Laws of Logarithms

   Solve for $x$, giving the answer to the nearest hundredth: $$5^{2x-1}=40.$$

10. Q10Writing Numbers

    Write each number in scientific notation, then order the three numbers from smallest to largest. Justify the ordering using the exponents rather than decimal expansions.

    1. $0.00042\times {10}^5$
    2. ${\displaystyle \frac{6\times {10}^{-3}}{2\times {10}^4}}$
    3. ${(3\times {10}^2)}^{-2}$
11. Q11Synthesis — drawing on several sheets in this topic

    The sound level of a machine is $L=10{\log }_{10}\left({\displaystyle \frac{I}{I_0}}\right)$ decibels, where $I$ is its intensity in $\text{W}/{\text{m}}^2$ and $I_0={10}^{-12}\text{W}/{\text{m}}^2$.

    In a workshop, a planer alone registers $78$ dB and a dust extractor alone registers $84$ dB. When two machines run together their intensities add.

    1. Determine the sound level of the workshop when both machines run, to the nearest tenth of a decibel.
    2. A trainee expects $78+84=162$ dB. Explain, using the laws of logarithms, why that reasoning is wrong.

The 8 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.

Absolute value is a distance, and that unlocks Q1

|a − b| is the distance between a and b on the number line. Nothing else about absolute value is worth memorising, because everything else follows. Q1 says a reading must differ from a target by at most a tolerance, which in that language is immediate:

Write the condition, unfold it into the double inequality, add c across all three parts, and the interval is the answer. Testing a specific reading then means computing the distance and comparing it with the tolerance — one line, with the absolute value bars actually written, because "it looks close enough" is not a justification.

Priority of operations, and the sign that is not part of the base

Q4 is an evaluation whose entire difficulty is one pair of brackets. It then asks you to say, in a sentence, why −2⁴ and (−2)⁴ differ.

Two other grouping symbols behave like brackets and are read the same way: a radical sign groups everything under it, and absolute value bars group everything between them. Simplify inside first, always. A cube root of a negative number is perfectly real — unlike an even root — which is why ∛(−27) is an ordinary −3 and not an error.

The laws of exponents, and the one habit that fixes most errors

Q7 is two simplifications. The laws themselves are short:

The habit: when a power is applied to a product, it lands on every factor, including the numerical coefficient. In (2a³b⁻²)⁴ the 2 becomes 16, and leaving it as 2 is the most frequent single mistake on questions of this shape.

For a fractional negative exponent on a fraction, do the three jobs in the easiest order: flip, root, power. Invert the fraction to kill the minus sign, take the root indicated by the denominator of the exponent, then raise to the numerator. Taking the root before the power keeps the numbers small — on (27/64)^(−2/3) that means working with 4 and 3 rather than with six-digit cubes.

Finally, "positive exponents only" is an instruction about the form of the answer, not about the algebra: finish the simplification with whatever signs come out, then move the negative-exponent factors across the fraction bar at the end.

Rationalizing: multiply by a well-chosen 1

Q2(a) and Q5 ask for a denominator with no radical in it. The tool depends on how many terms the denominator has.

A logarithm is an exponent — say it before every log question

Q3 asks for three logarithms without a calculator, and each one is answered by translating the notation back into a power:

So log₂(1/32) is the question "2 to what power gives 1/32?", and the negative exponent is the answer to that question. log₉27 asks the same of a base and a value that are both powers of 3, so the answer is a fraction — writing both numbers as powers of a common base is the move that makes it visible. A base smaller than 1 works identically; it simply forces the exponent negative.

Q8 uses the laws instead. There are three, and they mirror the exponent laws exactly, because a logarithm is an exponent:

Used left to right they break a logarithm apart; used right to left they collect several into one, which is what Q8 wants — condense to a single logarithm first, then evaluate it from the definition. A coefficient in front of a logarithm is a power hiding in plain sight: ½ log 81 is log √81.

Solving an exponential equation (Q9)

When both sides cannot be written as powers of the same base, take a logarithm of both sides and use the power law to bring the unknown down out of the exponent:

Then solve the ordinary equation that remains for x. Two practical points. Any base of logarithm works — base 10 and the natural logarithm give the same answer — because the base cancels in the quotient. And keep the quotient in exact form until the very last step: rounding the intermediate value and then multiplying by 2 doubles the rounding error, which is usually enough to lose the second decimal the question asks for.

Number sets, intervals, and scientific notation

Q6 wants the same set written two ways. Interval notation uses a square bracket at an included end and a reversed bracket at an excluded one, with infinity always reversed; set-builder notation names the set the values live in and the condition they satisfy, as in { m ∈ ℝ | 1.5 ≤ m < 8 }. Choosing the right set matters: a mass is a real number, so the interval is continuous, whereas intersecting an interval with ℤ produces a finite list you write out in extension. On that list, the brackets decide the two endpoints and nothing else does.

Q10 asks for scientific notation and then for an ordering justified by exponents. Scientific notation is a × 10ⁿ with 1 ≤ a < 10, and the normalising step is the one people skip: 0.00042 × 10⁵ is not yet in the form, because 0.00042 is not between 1 and 10. Once every number is normalised, the comparison is easy and is the point of the question — the power of 10 decides the order first, and the leading number only breaks ties between equal powers.

The synthesis question (Q11)

Decibels are defined by a logarithm, and Q11 asks what happens when two machines run at once. The physical fact given to you is that the intensities add, not the levels — so the method is: convert each level back to an intensity using the definition of a logarithm, add the intensities, then convert the total forward again. Part (b) then asks you to explain the trainee's wrong answer, and the explanation is the law that does not exist: adding two logarithms corresponds to multiplying what is inside them, which is not what adding intensities does. Factoring the smaller power of 10 out of the sum before taking the logarithm keeps the arithmetic clean.

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Getting the most out of it

Use Q2 as a diagnostic before anything else

It names four tools and asks which one each expression needs. Do it cold, first, and whichever tool you could not name is the section of this sheet to work through — that is more useful than working straight down the page.

No calculator on the logarithm questions

Q3 and Q8 are designed to be done from the definition and the laws. A calculator turns them into typing practice and hides exactly the skill the exam tests. Keep it for Q9, where a decimal answer is actually requested.

Stay in exact form until the final line

Radicals, fractions and logarithms all stay as they are until the question asks for a decimal. Rounding early and then continuing to compute is the most common way to produce an answer that is right in method and wrong in the last digit.

Common questions