Secondary 2 Exponents, Roots, Factors and Multiples

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. Q1Divisors, Multiples and Factorization

   The lockers in a Laval school are numbered starting at $1$.

   1. List all the divisors of $84$.
   2. List every multiple of $14$ between $50$ and $120$.
   3. Use the divisibility rules to say whether locker number $3150$ is divisible by $6$, by $9$, and by $4$. Give the rule you used each time.
   4. Name a locker number that is at the same time a divisor of $84$ and a multiple of $14$.
2. Q2Exponential Notation
   1. Write $7\times 7\times 7\times 7$ in exponential notation, then name the base and the exponent.
   2. Write $2^6$ in expanded form and give its value.
   3. Evaluate $3^4$, $5^3$, ${10}^4$ and $1^9$.
   4. A sheet of paper is folded in half over and over. Each fold doubles the number of layers, so after $n$ folds there are $2^n$ layers. How many layers are there after $7$ folds?
3. Q3Exponents
   1. Evaluate $2^5$, $3^3$, $6^2$ and ${10}^3$.
   2. Place $2^6$, $3^4$, $4^3$ and $5^2$ in order from smallest to largest. If two of them are equal, say so.
   3. In a video game, a player's score is $4^5$ points and a teammate's is $5^4$ points. Who has the higher score, and by how many points?
4. Q4Factoring a Number
   1. Break $540$ down into a product of prime factors, using a factor tree. Write the result in exponential notation.
   2. Do the same for $392$.
   3. One of $51$, $53$, $57$ and $91$ is prime. Which one? For each of the other three, give a factorization that proves it is not prime.
5. Q5Methods for Finding the LCM and the GCF Simultaneously

   Use the simultaneous division method: write the two numbers side by side and keep dividing by a prime that divides at least one of them, carrying down anything that does not divide.

   1. Apply the method to $84$ and $126$, then state the GCF and the LCM.
   2. Apply it to $45$ and $60$, then state the GCF and the LCM.
   3. Check one of your LCMs by showing it is a multiple of both starting numbers.
6. Q6Properties of Operations

   Compute each one mentally, then name the property of operations that made it easy.

   1. $25\times 17\times 4$
   2. $8\times 47+8\times 53$
   3. $6\times 98$
   4. $(36+59)+41$
7. Q7Square and Cubic Numbers
   1. List every perfect square between $100$ and $200$.
   2. Which of $216$, $300$ and $512$ are perfect cubes? Show your reasoning.
   3. A square section of a skatepark has an area of $169{\text{m}}^2$. Find the length of its side and its perimeter.
   4. A cube-shaped storage bin has a volume of $343{\text{cm}}^3$. Find the length of one edge.
8. Q8The Greatest Common Factor (GCF)
   1. Find the GCF of $48$ and $72$.
   2. Find the GCF of $54$, $90$ and $126$.
   3. A student council has $96$ muffins and $144$ clementines to share into identical snack bags, using everything up. What is the greatest number of bags they can make, and what goes into each bag?
9. Q9The Least Common Multiple (LCM)
   1. Find the LCM of $12$ and $18$.
   2. Find the LCM of $8$, $12$ and $20$.
   3. At a market, one supplier delivers every $9$ days and another every $12$ days. Both delivered today. In how many days will they next deliver on the same day, and how many times will that happen over the next $180$ days?
10. Q10The Root of a Number | Secondary 1 and 2
    1. Evaluate $\sqrt{196}$, $\sqrt{625}$, $\sqrt[3]{125}$ and $\sqrt[3]{1000}$.
    2. A square banner has an area of $324{\text{cm}}^2$. How long is each side?
    3. A cube-shaped crate has a volume of $512{\text{cm}}^3$. Find the length of an edge, then find the total area of its six faces.
11. Q11Synthesis — drawing on several sheets in this topic

    Two numbers are given: $360$ and $84$.

    1. Write each as a product of prime factors, in exponential notation.
    2. Use those factorizations to find the GCF and the LCM of the two numbers.
    3. Check your two answers using the relationship GCF $\times$ LCM $=$ the product of the two numbers.

The 11 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.

Almost every question here is the prime factorization in disguise

Divisors, the GCF, the LCM, perfect squares, perfect cubes — they look like six separate topics and they are one. Break a number into primes and each of those questions becomes a matter of counting how many copies of each prime you are holding. Students who learn the six methods separately spend the year guessing which to use; students who learn to factor first spend it deciding what to do with the factors.

Divisors and multiples point in opposite directions (Q1)

The two words get swapped constantly, and the fix is to notice that a list of divisors is finite while a list of multiples never ends. The divisors of 84 stop at 84; the multiples of 14 run on forever, which is why Q1 has to fence them in with "between 50 and 120".

Find divisors in pairs and you will not miss one. Start at 1 and walk upwards, writing each partner beside it — 1 with 84, 2 with 42, 3 with 28, and so on — and stop when the two columns meet in the middle. Listing them one at a time from memory is how a divisor goes missing.

Exponential notation: the exponent counts, it does not multiply (Q2, Q3)

In a power, the base is the factor being repeated and the exponent is how many times it appears. So four cubed is 4 × 4 × 4 = 64, and the single most common error on this topic — reading it as 4 × 3 = 12 — is not an arithmetic slip but a misread of what the small number is for. Say the meaning out loud once per question and the error stops happening.

Q3 then asks you to order several powers, and the point of it is that you cannot compare powers by looking at them. A bigger base does not mean a bigger value, and neither does a bigger exponent: 2⁶ and 4³ are both 64, while 4⁵ comfortably beats 5⁴. Work each one out and compare the values — the ordering is regularly the opposite of what the notation suggests.

GCF or LCM? Read the situation, not the numbers (Q8, Q9)

Both questions hand you two numbers and a story, and choosing the wrong one of the two methods gives a clean, confident, wrong answer. The deciding question is whether the thing you are looking for is smaller than what you started with or bigger.

The ladder that gives you both at once (Q5)

Write the two numbers side by side and keep dividing by a prime, carrying down anything the prime does not divide. When the row reaches 1 and 1, read the ladder twice: the primes that divided both numbers multiply to the GCF, and every prime you used multiplies to the LCM. One piece of work, two answers.

The step people get wrong is the carrying down. Once a prime divides only one of the two numbers, the other is copied unchanged into the next row — it is not divided, and it is not dropped. Keep the two columns lined up on the page and mark the rows where both numbers went, because those marks are exactly your GCF.

Squares, cubes, and reading them backwards (Q7, Q10)

A perfect square is a number that can be written as a whole number times itself; a perfect cube, as a whole number times itself three times. Knowing the squares to about 15 and the cubes to about 10 by heart turns most of this sheet into recall, and it is a genuinely small list.

The geometry questions are those same facts read from the other end. A square section with an area of 169 m² has a side of 13 m because 13 × 13 = 169; a cube-shaped bin holding 343 cm³ has an edge of 7 cm because 7 × 7 × 7 = 343. Carry the units through: a length in centimetres, an area in square centimetres, a volume in cubic centimetres — the exponent in the unit is telling you which root you need.

The properties of operations are shortcuts, not vocabulary (Q6)

Q6 asks for the answer and the property that made it easy, and the second half is the point. Commutativity and associativity let you reorder and regroup — 25 × 17 × 4 becomes 25 × 4 × 17, which is 100 × 17 and needs no written working at all. Distributivity runs in both directions: forwards it turns 6 × 98 into 6 × 100 − 6 × 2, and backwards it collapses 8 × 47 + 8 × 53 into 8 × 100.

Worth knowing where it stops: subtraction and division have neither commutativity nor associativity, and one counterexample settles that permanently. 7 − 3 and 3 − 7 are different numbers, and so are (12 − 5) − 3 and 12 − (5 − 3).

The synthesis question, and the check it teaches (Q11)

The last free question factors two numbers, takes the GCF and the LCM straight from those factorizations, and then verifies both at once with a relationship worth carrying for the rest of school: for any two whole numbers, the GCF multiplied by the LCM equals the product of the two numbers. It costs one line, it catches a mis-taken exponent immediately, and if you ever know one of the two it hands you the other without listing anything.

Getting the most out of it

Factor first, decide second

When a question involves two numbers and a common something, write both prime factorizations before you decide what is being asked. With the factorizations on the page, the GCF and the LCM are each one line of reading — shared primes at the smaller count, all primes at the larger count — and the choice between them stops being a guess.

Say what the exponent means, every time

Out loud, or under your breath: "five cubed is five times five times five." It takes two seconds and it is the only reliable cure for the error that turns a power into a multiplication.

Check the size of the answer before you check the arithmetic

A GCF is never larger than the smaller number; an LCM is never smaller than the larger one; the square root of a number above 1 is smaller than the number itself. These bounds cost nothing to apply and they catch the mistakes that a second run through the same arithmetic would only repeat.

Common questions