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Secondary 3 Exponents, Roots and Scientific Notation Worksheet

By Marius — private math tutor in Montreal. Updated 30 August 2026

Secondary 3 turns exponents from shorthand into a small algebra of their own. Five laws replace counting factors by hand, an exponent of zero and a negative exponent both acquire meanings you cannot guess from repeated multiplication, roots come back as the operation that undoes a power, and the whole apparatus gets pointed at scientific notation — the way every large or small measurement is written from here on. All of it is designed to be done exactly, without a calculator. Read it here, and print the PDF when you want to write on it; it costs nothing and asks for no account.

Practice worksheet — free PDF

5 pages 8 questions Letter size, print-ready

Download the worksheet Read the questions here

No email, no account, no watermark. Teachers: photocopy it for your classes freely. Worked solutions and 7 harder problems come with the Secondary 3 Math bundle.

All 8 questions

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

Q1Exponential Notation

In parts (a) to (c), rewrite the product in exponential notation, then evaluate it. Part (d) asks for something different.
1. $7 \times 7 \times 7 \times 7$
2. $(- 4) \times (- 4) \times (- 4)$
3. $\frac{2}{5} \times \frac{2}{5} \times \frac{2}{5}$
4. Evaluate $(- 2)^{6}$ and $- 2^{6}$ . The two answers are different — say what the exponent is applied to in each case.
Q2Exponents –

Evaluate each power without a calculator. Give exact answers as fractions where they are not whole numbers.
1. $9^{0}$
2. $2^{- 5}$
3. $(- 3)^{- 2}$
4. ${(\frac{3}{4})}^{- 2}$
5. $10^{- 3}$ , written both as a fraction and in decimal notation
Q3Operations

The priority rules decide the order in which operations are carried out.
1. Evaluate $\sqrt{81} + 2 \times (7 - 3)^{2} - 5^{0}$ , writing one line per step and naming the operation you did at each step.
2. Marc claims that $2 \times (7 - 3)^{2} = (2 \times 4)^{2} = 64$ . Which priority rule has he broken? Correct his step.
Q4Scientific Notation

A class prepares a science-fair poster and must put every measurement into the same form.
1. Write in scientific notation: $4 850 000$ ; $0.000 062$ ; $903$ .
2. Write in ordinary decimal notation: $7.4 \times 10^{- 4}$ ; $1.06 \times 10^{7}$ .
3. One student writes a length as $38 \times 10^{5} m$ . Explain why this is not scientific notation, and repair it.
Q5Square and Cubic Numbers –
1. A square patio has an area of $196 m^{2}$ . Find the length of one side.
2. A cubic recycling bin has a volume of $1728 {cm}^{3}$ . Find the length of one edge.
3. From the list $64$ , $100$ , $125$ , $216$ , $400$ , say which numbers are perfect squares, which are perfect cubes, and which are both.
Q6The Laws of Exponents

Simplify each expression using the laws of exponents. Leave every answer with positive exponents only.
1. $x^{5} \cdot x^{7}$
2. $\frac{a^{9}}{a^{4}}$
3. ${(y^{3})}^{4}$
4. ${(2 m^{4})}^{3}$
5. $\frac{12 b^{6}}{4 b^{6}}$
Q7The Root of a Number \textbar{} Secondary 3

Evaluate each root exactly.
1. $\sqrt{225}$
2. $\sqrt[3]{- 64}$
3. $\sqrt[4]{81}$
4. $\sqrt{0.49}$
5. $\sqrt[3]{\frac{27}{125}}$
Q8Synthesis — drawing on several sheets in this topic

A radio signal travels at $3 \times 10^{8} m / s$ . A ground station sends a signal to a drone $1.2 \times 10^{4} m$ away.
1. How long does the signal take to reach the drone? Give the answer in scientific notation, and name the law of exponents you used on the powers of ten.
2. How far does such a signal travel in $2 \times 10^{- 3} s$ ? Give the answer in scientific notation.

The 7 challenge problems for this topic are a separate, paid sheet and are not reproduced here.

Which stream is this for? Secondary 3 is the common year: everyone takes the same mathematics before the CST, TS and SN options separate in Secondary 4. The laws practised here are used constantly by all three, and they are also the sharpest early signal on the sheet — a student who can apply them without hesitating finds next year's algebra much lighter work.

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 what the exponent is attached to

Q1 ends with two expressions that look almost identical and are not equal, and getting the difference straight now prevents a long run of sign errors later.

(−2)⁶ = 64 but −2⁶ = −64

The brackets decide the base. In the first, the base is −2 and six of them are multiplied together, so the negatives pair off and the answer is positive. In the second, only 2 is raised to the sixth power and the minus sign is applied afterwards — a power outranks a sign in the order of operations. Same digits, different question.

The sign of a power of a negative base follows the parity of the exponent. An even exponent pairs every negative with another and gives a positive result; an odd exponent leaves one over. That is enough to give the sign of a power far too large to evaluate, which is exactly what the challenge question asks for — and to explain why an expression with no brackets does not obey the rule at all.

Zero and negative exponents mean something you cannot count out

Q2 evaluates powers exactly, including a zero exponent, negative exponents on whole numbers and on a fraction. "Multiply the base by itself zero times" is not a sentence with a meaning; the definitions come from making the division law keep working.

a⁰ = 1 and a⁻ⁿ = 1 ÷ aⁿ

A negative exponent is an instruction to flip, not a sign. 2⁻⁵ is one thirty-second — a small positive number — and never −32. On a fraction the flip is the whole operation: (¾)⁻² becomes (4/3)², which is 16/9.

And a negative exponent turns comparisons upside down. The challenge question is built on this: with a positive exponent the bigger base gives the bigger power, and with a negative exponent it gives the smaller one, because you end up dividing by a bigger number. The conclusion drawn from three examples is true and incomplete, and repairing it means naming the condition the examples all happened to satisfy.

The five laws, and the one place they are misread

Q6 applies all of them: multiplying powers of the same base, dividing them, raising a power to a power, raising a product to a power, and a quotient that comes out to a zero exponent.

What each law does to the exponents

Every one of them requires the same base. That condition is doing more work than the arithmetic.

1
Multiplying: add the exponents
x⁵ · x⁷ = x¹². Two lots of factors are being counted, so the counts add.
2
Dividing: subtract them
a⁹ ÷ a⁴ = a⁵. Push the subtraction one step further than seems reasonable and you get the zero exponent and the negative one for free — which is where those definitions came from.
3
A power of a power: multiply them
(y³)⁴ = y¹². This is the one confused with the first law, and the confusion runs both ways.
4
A power of a product: it reaches every factor
(2m⁴)³ = 8m¹² — the coefficient is a factor and gets cubed too. Leaving the 2 alone is the most common single error in the topic.
5
The coefficients are ordinary arithmetic
12b⁶ ÷ 4b⁶ is 3. The numbers divide, the letters follow their own law, and the two are not mixed.

The challenge question is a worked solution containing two errors, and you have to name the law misused in each. One step raises a product to a power and forgets the coefficient; another multiplies two powers and multiplies the exponents instead of adding them. Both are the ordinary errors, and finding them in someone else's work is much harder than avoiding them in your own — which is why the question is worth doing slowly.

Priority rules, with a power in the mix

Q3 evaluates an expression containing a root, a bracket, a power and a zero exponent, one line per step, naming the operation each time. It then shows a student who multiplied before applying the exponent, and asks which rule was broken.

An exponent binds only to what it is written on. In 2 × (7 − 3)² the bracket is squared and the 2 is not, so the value is 2 × 16 = 32. Multiplying first to get (2 × 4)² squares the 2 as well, and that is a different expression. Naming the rule is the answer here — showing the correct chain of arithmetic without saying what went wrong is half a response.

Roots undo powers, and the factor tree tells you which ones come out

Q7 evaluates square, cube and fourth roots exactly, including a root of a decimal and a root of a fraction — and a cube root of a negative number, which exists, unlike the square root of a negative one. Q5 works with perfect squares and perfect cubes directly, sorting a list into which is which and which is both.

The challenge question behind those is the prime factorisation one, and it is the most useful idea on the sheet. Write 720 as a product of primes with exponents, and you can read off immediately what is missing: a number is a perfect square exactly when every exponent in its factorisation is even, and a perfect cube exactly when every exponent is a multiple of three. Finding the smallest multiplier that repairs the exponents is then bookkeeping rather than searching.

Roots scale differently from the quantities they measure. The other challenge question makes the point with crates: multiplying a cube's volume by 8 multiplies its edge by 2, and multiplying the volume by 27 multiplies the edge by 3, because the cube root of the factor is what reaches the edge. Asked which factor makes the edge ten times longer, the answer is a thousand — and the question specifically asks you to argue it from the cube root rather than by trying numbers until one works.

Scientific notation: one digit before the point, and a power of ten

Q4 converts in both directions and then hands you a number that is not in scientific notation at all, and asks you to repair it. That third part is the one worth attention.

The mantissa must be at least 1 and less than 10. 38 × 10⁵ is a perfectly correct value and is not scientific notation, because 38 has two digits before the point. Rewriting 38 as 3.8 × 10 and absorbing that extra ten into the power gives 3.8 × 10⁶. Every digit moved past the point is paid for by one step in the exponent — moving the point left raises the exponent and moving it right lowers it.

The same repair is what makes the challenge question about multiplication work. Multiply 5 × 10⁶ by 8 × 10⁴ and the mantissas give 40, which has to be renormalised — so the exponent of the answer is one more than the sum of the two exponents. The claim that the exponents simply add is true only when the product of the mantissas stays below 10, and stating that condition exactly is what the question is after.

The synthesis questions put it all together on measurements: a signal travelling at a given speed across a given distance, which is a division of powers of ten; and a cube-shaped container whose volume is given in scientific notation, where the cube root only comes out cleanly after the exponent has been rewritten as a multiple of three. That last step is the whole question — the exponent has to be divisible by 3 for the root to land on a whole power of ten, so the mantissa is adjusted until it is.

Getting the most out of it

Put the calculator away

Every question here is built to come out exactly: the roots are whole, the powers are small, the scientific notation is clean. A calculator hides whether you know the law, and it will not help with the questions that ask you to name one.

Say the law out loud before you use it

"Same base, multiplying, so the exponents add." It sounds laborious for about a day and then stops being necessary. The step it prevents is the one where a power of a power gets its exponents added instead of multiplied — the two laws are only distinguishable by which situation you are in.

Factor into primes whenever a question mentions squares or cubes

The exponents in the prime factorisation answer the question directly: all even means a perfect square, all multiples of three means a perfect cube, and whatever is missing from that pattern is exactly what has to be multiplied in.

Check the mantissa every time you write scientific notation

At least 1, less than 10. Most scientific-notation errors are not arithmetic — they are answers left in a form that is numerically right and not in the required notation, and the fix is one step of the decimal point paid for in the exponent.

Want the solutions, or something more challenging?

The worksheet, the questions above and every explanation on this page stay free permanently. Three more PDFs exist for this topic — the worked answer key, a harder problem set, and the answer key to that. They come with the Secondary 3 Math Solutions Bundle, which is what keeps the rest of the series free.

What else exists for Exponents, Roots and Scientific Notation

Three PDFs · 10 pages · all three are in the bundle below.

Answer key — 2 pages. All 8 questions worked step by step, including the restrictions and the justifications. Not a list of final answers.
Challenge problems — 5 pages, 7 problems. A separate sheet at exam-plus difficulty covering the same 7 concepts. Harder than anything on the free sheet.
Challenge answer key — 3 pages. Every challenge problem worked to the same standard, with the checks shown.
PDF, letter size, print-ready.

In the bundle See what's in it Not sold separately

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One download, one payment, the whole program. Every answer key and every challenge set for all 11 Secondary 3 Math worksheet sets — including this one.

11 sets · 33 PDFs · 154 pages$19.99

Worked solutions, not answer lists — every step written out
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Common questions

Is this worksheet really free?

Yes — the questions are on this page to read, and the PDF downloads directly, no email and no account. The one paid item is optional: the complete Secondary 3 Solutions Bundle, which covers every set at this level.

What do I get if I buy the bundle?

Every answer key and every challenge set for the whole Secondary 3 series, in one download. For this topic that is the worked answer key to the practice questions, a set of harder problems, and the worked answer key to those. The practice worksheets themselves stay free.

Does a negative exponent make the answer negative?

No. A negative exponent means take the reciprocal, so 2 to the power −5 is one thirty-second, a small positive number. On a fraction it flips the fraction over before the power is applied, which is usually the easiest order to work in.

Why is (−2) to the sixth different from −2 to the sixth?

Because the brackets decide what the exponent is attached to. In the first, the base is −2 and six of them multiply together, so the negatives pair off and the result is positive. In the second, only 2 is raised to the power and the minus sign is applied afterwards, since a power outranks a sign in the order of operations.

When do exponents add and when do they multiply?

They add when two powers of the same base are multiplied together, and they multiply when a power is itself raised to a power. Saying which situation you are in before you write anything is what keeps the two apart — they are the pair that gets confused most often, in both directions.

What counts as proper scientific notation?

A single non-zero digit before the decimal point — a mantissa at least 1 and less than 10 — times a power of ten. A number like 38 × 10⁵ has the right value and the wrong form; moving the point one place left and raising the exponent by one repairs it.

Which Secondary 3 stream is this for?

There is no stream yet. Secondary 3 is the common year every student takes before the CST, TS and SN options separate in Secondary 4, and the laws of exponents are used constantly by all three. This is one of the best early indicators of how comfortable next year will feel.

Can teachers use this in class?

Yes. Print and photocopy it for your own classes freely — I just ask that the tutorinmontreal.ca footer stays on the page.

I'm stuck on one question. Can you help?

Yes — through one-on-one tutoring, in Montreal or online. Get in touch to arrange a session, or see the current rates.

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