Mini Minds Hub · Free printable
Indices and Powers: Free Printable Worksheet for Year 7
- ForYear 7
- StageKey Stage 3
- CostFree
- FormatPrintable PDF
Indices and Powers is a free printable maths worksheet for Year 7 (ages 11 to 12), in Key Stage 3. Children use the index laws to simplify powers of numbers and of letters, and they work out some powers as plain numbers. It needs only a pencil, and page 2 gives the grown-up notes, help and stretch ideas, and the answers.
It comes from the Mini Minds Hub, the free library of printables from Jaques of London, the British games and toy company that has been making games for families since 1795. Print both pages, give page 1 to the child and keep page 2 for yourself. Black and white printing works well.
What is on the sheet
Children meet numbered questions with space to write each answer. Some ask for a missing power, so children must decide what goes in the box. Some use a letter instead of a number, which shows that the laws work for any base. A few ask children to work out a power as an ordinary number. The questions cover multiplying, dividing and raising a power to a power.
- 3^2 x 3^4 = 3^?
- 5^7 ÷ 5^3 = 5^?
- (2^3)^2 = 2^?
- 7^0 = ?
- a^3 x a^5 = ?
- Work out 2^5
- Work out 10^3
- y^9 ÷ y^4 = ?
A worked example
Simplify 6^8 x 6^6, giving your answer as a single power.
This question is different from the ones on the sheet. It shows two ways to think about the first index law.
Method 1: write it out in words- 6^8 means eight sixes multiplied together.
- 6^6 means six sixes multiplied together.
- Multiply them and you have 8 + 6 sixes in one long multiplication.
- 8 + 6 = 14, so there are 14 sixes. That is 6^14.
- Check that the bases match. Both are 6.
- Keep the base the same.
- Add the powers: 8 + 6 = 14.
- Write the answer as 6^14.
What are indices and powers?
An index tells you how many times a number is used in a multiplication. In 8^4, the 8 is the base and the 4 is the index, also called the power. The plural of index is indices. Writing repeated multiplication this way is shorter and easier to read, especially when the same number or letter appears many times.
The index laws only work when the bases match. With the same base, multiplying adds the powers, dividing takes one power away from the other, and a power of a power multiplies the powers. Letters follow the same laws as numbers, because a letter stands for any number. Writing the factors out shows why each law works and helps children remember it.
Skills and curriculum link
- Index laws
- Simplifying
- Powers
- Algebraic notation
- Spotting patterns
- Checking answers
Ideas to help you explain the laws
- Base and indexThe base is the number or letter being multiplied. The index is the small raised number that says how many times it is used.
- Multiplying with the same baseEach power counts how many copies of the base there are. Multiplying joins the two groups of copies into one bigger group.
- Dividing with the same baseDividing cancels copies of the base from the top and bottom. What is left is the difference between the two powers.
- A power of a powerA power of a power means a group of copies, repeated several times. Children can write it out in full to see how the powers combine.
Key words
- IndexThe small raised number that shows how many times the base is multiplied by itself. The plural is indices.
- BaseThe number or letter that is being multiplied by itself.
- PowerAnother word for index. It can also mean the whole thing, so 4^3 is a power of 4.
- SimplifyWrite something in a shorter or neater form without changing its value.
- ProductThe answer when you multiply.
- VariableA letter that stands for a number.
How to use it
- Print both pages. Page 1 is for the child. Keep page 2, which has the notes and answers.
- Check that the child has been taught the three index laws. If not, go over them first with a short example.
- Do the first question together. Ask the child to say which law they are using and why.
- Let the child work alone, then check against the answers on page 2. Ask, "How did you know which law to use?"
- Come back another day and redo any questions that felt tricky. Repeating a familiar sheet builds confidence.
Help, stretch and game ideas
If it feels too hard
Write a power out as a long multiplication. For example, 4^2 x 4^3 is 4 x 4 x 4 x 4 x 4. Count the fours. Do this for a few questions until the pattern in the powers is clear, then try without writing it out.
To stretch confident children
Work out a rule for what happens to the powers in (a^m)^n and test it with numbers of your own. Then make up two questions for a partner to simplify. Check their answers using your rule.
Make it a game
Roll a die three times. The first roll is the base. The second and third rolls are two powers of that base. Write the multiplication, then simplify it to a single power. Swap with a partner and check each other's work.
Mistakes to watch for
Watch forA child multiplies the bases as well, so 9^4 x 9^3 becomes 81^7.
Try thisKeep the base the same. Write the nines out. The base only tells you what is being multiplied. The powers count how many.
Watch forA child reads 6^2 as 6 x 2 and writes 12.
Try thisSay it as "two sixes multiplied together" and write 6 x 6. The index counts copies of the base.
Watch forA child adds the powers in (4^2)^5 and writes 4^7.
Try thisWrite it out: (4^2)^5 is five groups of 4^2. Count how many fours that gives in total, then find the quicker rule.
Talk about it
- Which index law did you use for that question, and how did you choose?
- Why do the bases have to be the same before we use the laws?
- How could you check an answer by writing the powers out in full?
- What is the difference between 5 x 3 and 5^3?
- Does a letter follow the same laws as a number? Why?
- Can you make up a question where a tempting wrong answer is easy to make?
Take it off the page
Long multiplication match
Write ten powers on small cards and their long multiplication forms on others. Mix them up and match the pairs, then say how many copies of the base each one has.
Law posters
Make a poster for each index law with a title, a rule in words and two examples of your own. Use a base that is not on the sheet.
Pattern table
Pick a base and write a table of its first few powers by multiplying each answer by the base again. Talk about the pattern you can see in the powers.
Who it is for
Parents
Sit beside your child for the first question, then let them lead. Ask them to explain the law they are using in their own words. If they are stuck, ask them to write the power out as a long multiplication.
Teachers
Use it as independent practice after teaching the three index laws, as a starter or plenary, or as homework. Pairs can check each other's work. Page 2 has support and stretch ideas for mixed ability groups, and the answers.
Home educators
Use it after you have covered the index laws in a wider algebra plan. Talk through each law with a short example first. The sheet can be printed again for brothers and sisters or used for revision a few weeks later.
Where indices fit in maths
Before this, children need secure times tables and a feel for squares and cubes. They also need to understand that a letter can stand for a number and that repeated multiplication can be written more briefly.
After this, children use indices in expanding and simplifying algebraic expressions and in standard form. Later work in Key Stage 4 builds on the same laws, so a firm grasp now makes that work easier.
Questions parents and teachers ask
- What age is Indices and Powers for?
It is written for Year 7 (ages 11 to 12), in Key Stage 3. It also works as revision for older pupils who need to firm up the index laws, or as extension for confident younger ones.
- What do children need to know first?
They should know that a small raised number shows repeated multiplication. It helps if they have been taught the three index laws for multiplying, dividing and raising a power to a power before they start.
- Is this worksheet free?
Yes. It is free to print for home and classroom use. Print both pages, give page 1 to the child and keep page 2 for the grown-up.
- Is it a screen free activity?
Yes. It needs only paper and a pencil. It suits quiet time after school, a rainy afternoon or a holiday, and it gives eyes a rest from screens.
- What if my child mixes up the laws?
Go back to writing powers out as long multiplications. Counting the copies of the base shows why the powers are added, taken away or multiplied. Then try the question again without writing it out.
- Can teachers use it in class?
Yes. It works as a starter, plenary, early finisher task, homework or table activity. It needs no preparation beyond printing, and page 2 gives support and stretch ideas for mixed ability groups.
- Where are the answers?
The answers are on page 2, which is for the grown-up. Check them together after the child has finished and ask them to explain how they worked each one out.
Preview the full PDF
Made by Jaques of London, making games for families since 1795. A printable supports classroom learning and does not replace a teacher.