aⁿ Notation
Products of equal factors and powers, base and exponent, saying a power (squared, cubed, to the power of) and a number as a power of a given base.
Start over?
Teaching guide: powers and square roots
Powers are a short way of writing a product of equal factors, and the square root is the operation that undoes squaring. In Grade 6 pupils meet squares, cubes and powers of 10; in Grade 7, the laws of exponents and exact and whole-number square roots; in Grade 8, negative exponents and standard form. This free generator creates A4 worksheets for every step, with an answer key that shows the working behind each answer.
Student worksheet 
Answer key 3² Working out
Squares and cubes, powers of any base, the exponents 0 and 1, and powers of decimals and fractions.
10ⁿ Powers of 10
1 followed by zeros, multiplying by powers of 10, expanded form and standard form for large numbers.
aᵐ×aⁿ Laws of exponents
Multiplying and dividing powers of the same base, a power of a power, the same exponent, a⁰ = 1 and negative exponents, with every step in the answer key.
Square roots
Square numbers, exact roots, roots from prime factors, whole-number roots with a remainder by trial or the long method, and estimating roots between whole numbers and tenths.
Word problems
The side of a square garden, the volume of a box, chain messages, bacteria that split and pupils in a square formation.
Frequently asked questions about powers and square roots worksheets
What is a power and how do you say it?
A short way of writing a product of equal factors: 2 × 2 × 2 × 2 × 2 = 2⁵. The base (2) is the repeated factor and the exponent (5) says how many times it is repeated. You say «two to the power of five»; with exponent 2 you say «squared» and with exponent 3, «cubed».
What are the laws of exponents?
With the same base, multiplying adds the exponents (2³ × 2⁴ = 2⁷) and dividing subtracts them (2⁷ ÷ 2³ = 2⁴). A power of a power multiplies them ((2³)² = 2⁶). With the same exponent you multiply or divide the bases (2⁵ × 3⁵ = 6⁵). Any power with exponent 0 equals 1, and a⁻ⁿ = 1/aⁿ.
Why is a⁰ = 1?
Because aⁿ ÷ aⁿ = 1 (a number divided by itself) and, by the division law, aⁿ ÷ aⁿ = aⁿ⁻ⁿ = a⁰. Level 2 of the laws of exponents uses this so pupils can see it.
How do you find a whole-number square root?
Find the two square numbers the number lies between: for 70, 8² = 64 < 70 < 81 = 9². The whole-number root is 8 and the remainder is 70 − 64 = 6, so 70 = 8² + 6. For large numbers there is the long method for square roots, also in the column operations worksheets.
How do you write a number in standard form?
As a number from 1 to less than 10 multiplied by a power of 10: 45,000,000 = 4.5 × 10⁷. The exponent is the number of places the decimal point moves. Level 3 of powers of 10 practises both directions.
Which grades are these worksheets for?
Squares, cubes, powers of 10 and the first square roots are for Grade 6; the laws of exponents, exact and whole-number roots and expanded form for Grade 7; negative exponents, powers of fractions and standard form for Grade 8. Powers with a negative base and the sign rules are in the integer worksheets.
Which other worksheets go well with these?
The order of operations, which mixes powers and roots with brackets, prime factorisation and integers. You will find every resource in math worksheets.
How do I print only the answer key?
Press Print and choose “Answer key”, or “Worksheet and answer key” for both. You can also save it as a PDF from the browser's print dialog. The worksheets are made in your browser and no data is sent.
More worksheet generators you may like
Times Tables
A4 worksheets for the tables you choose: complete the table, shuffled facts, missing factors, multiplication grids and division facts, in three levels with an answer key.
Math Operations
Generate high-density A4 printable worksheets (15-20 problems per sheet) with regrouping addition, borrowing subtraction, multi-digit multiplication, long division, and square roots.
Mental Maths Drills
Drills of 50 to 100 quick facts in line with time and score boxes: addition, subtraction, number bonds to 10 and 100, ± 10, doubles and halves, times tables and × 10, 100 and 1000.