Rational numbers, exponents and roots
Classifying rational and irrational numbers, the laws of exponents, and square and cube roots
About four lessons of 45 to 60 minutes
Not every number can be written as a fraction, and that is a big deal
Every whole number, fraction and repeating decimal you have ever used can be written as one whole number divided by another: a rational number. But some numbers, like the exact diagonal of a 1-by-1 square, can never be written that way, no matter how hard you try. Ancient mathematicians found this so unsettling that (according to legend) the discovery was kept secret. Today it just has a name: irrational.
Exponents and roots are the tools that keep turning up alongside this idea: a square root can land you on a rational number (sqrt 49 = 7) or an irrational one (sqrt 2 has no exact fraction), and the same shortcut rules for multiplying and dividing powers make huge expressions manageable without a calculator.
- The diagonal of a 1x1 squareis exactly sqrt(2), a number that is never a whole-number fraction
- A repeating decimal like 0.777...always converts back to an exact fraction, 7/9
- A computer's memory in bytesis described with powers of 2, like 2^10 = 1024
- A cube-shaped box's edge lengthfound from its volume with a cube root
What students will be able to do
Students will classify numbers as rational or irrational, convert a repeating decimal to an exact fraction, apply the properties of integer exponents to simplify numerical expressions, and evaluate square roots and cube roots of small perfect squares and cubes.
- I can explain the difference between a rational number and an irrational number.
- I can convert a repeating decimal, such as 0.777..., into an exact fraction.
- I can simplify an expression using the product, quotient and power rules for exponents.
- I can evaluate a zero exponent and a negative exponent.
- I can evaluate square roots of perfect squares and cube roots of perfect cubes, including negative cubes.
Standards this unit teaches
- 8.NS.A.1Common Core (US)Rational and irrational numbers
Understand that numbers are either rational or irrational and convert repeating decimals to fractions.
- 8.EE.A.1Common Core (US)Properties of exponents
Apply the properties of integer exponents to write equivalent numerical expressions.
- 8.EE.A.2Common Core (US)Square roots and cube roots
Use square root and cube root symbols to represent solutions and evaluate roots of small perfect squares and cubes.
Prior knowledge
This unit builds on skills students should already have met. Revisit any that are shaky first.
- Rational numbers and the number linefractions and decimals as points on a number line, the foundation for classifying numbers
- Square numbers and square rootsthe Grade 7 groundwork this unit's root work builds on
- Order of operationsexponents are evaluated before multiplication and addition, exactly as PEMDAS says
Words to teach and display
- Rational number
- a number that can be written as one integer divided by another (a fraction), including terminating and repeating decimals
- Irrational number
- a number that can never be written as an exact fraction; its decimal never ends and never repeats
- Exponent
- the small raised number that tells you how many times to multiply the base by itself
- Square root
- a number that, multiplied by itself, gives the original number, written with the symbol sqrt
- Cube root
- a number that, multiplied by itself three times, gives the original number
- Perfect square / perfect cube
- a number that is exactly the square (or cube) of a whole number, such as 49 (7^2) or 27 (3^3)
Teach it: concrete, pictorial, abstract
The lesson moves from things students can hold, to pictures and diagrams, to the written maths. The diagrams below are drawn from data, so they are accurate and print cleanly. Teach straight from them.
1. Rational vs irrational numbers
ConcreteA rational number is any number you can write as one whole number divided by another: 3/4, 7 (which is 7/1), 0.5 (which is 1/2), and even a repeating decimal like 0.333... (which is 1/3). An irrational number can never be written that way; its decimal goes on forever without ever settling into a repeating pattern.
Square roots are the most common source of irrational numbers in Grade 8: sqrt(49) = 7 is rational (49 is a perfect square), but sqrt(2) is irrational, because no fraction, however precise, squares to exactly 2. You will meet sqrt(2) again as the diagonal of a unit square, and it shows up throughout geometry.
- Is 0.5 rational or irrational, and why?
- Why is sqrt(2) irrational but sqrt(4) rational?
2. Converting a repeating decimal to a fraction
PictorialEvery repeating decimal is secretly a fraction. The trick is to multiply by a power of 10 that shifts the repeating block, then subtract the original to cancel the repeating part entirely.
For x = 0.777... (the 7 repeats forever), multiplying by 10 gives 10x = 7.777.... Subtracting the original equation cancels every repeating digit: 10x - x = 7.777... - 0.777... = 7, so 9x = 7 and x = 7/9.
Convert the repeating decimal 0.777... to an exact fraction.
- Let x = 0.777....
- Multiply both sides by 10 (one repeating digit): 10x = 7.777....
- Subtract the original equation: 10x - x = 7.777... - 0.777..., so 9x = 7.
- Divide both sides by 9: x = 7/9.
Answer: 0.777... = 7/9.
- Why does subtracting the two equations cancel the repeating digits?
- If a decimal repeats a two-digit block, what would you multiply by instead of 10?
3. The properties of exponents
AbstractThree rules let you simplify expressions with exponents without ever expanding them out: multiplying same-base powers ADDS the exponents, dividing same-base powers SUBTRACTS them, and raising a power to a power MULTIPLIES them.
Product rule: a^m x a^n = a^(m+n). For example, 2^3 x 2^4 = 2^7 = 128 (check: 2^3 = 8 and 2^4 = 16, and 8 x 16 = 128). Quotient rule: a^m / a^n = a^(m-n). Power rule: (a^m)^n = a^(mn). A zero exponent always gives 1 (a^0 = 1), and a negative exponent flips the base into a fraction: a^-n = 1/a^n, for example 4^-2 = 1/16.
Simplify (2^3 x 2^5) / 2^4 using the properties of exponents.
- Apply the product rule in the numerator: 2^3 x 2^5 = 2^(3+5) = 2^8.
- Apply the quotient rule: 2^8 / 2^4 = 2^(8-4) = 2^4.
- Evaluate: 2^4 = 16.
Answer: (2^3 x 2^5) / 2^4 = 16.
- Why do you ADD exponents when multiplying same-base powers, but SUBTRACT when dividing?
- What does 4^-2 equal, and why is it not a negative number?
4. Square roots and cube roots
AbstractA square root undoes squaring: sqrt(49) = 7 because 7^2 = 49. A cube root undoes cubing: the cube root of 27 is 3 because 3^3 = 27. Unlike square roots, cube roots of negative numbers are perfectly normal: the cube root of -8 is -2, because (-2)^3 = -8.
Solving x^2 = 25 has two solutions, x = 5 or x = -5, because both 5^2 and (-5)^2 equal 25. Solving x^3 = 64 has only one real solution, x = 4, since cubing keeps the sign of the original number (a negative number cubed stays negative).
Solve x^2 = 81 and x^3 = 64.
- For x^2 = 81: find a number that squares to 81. 9^2 = 81 and (-9)^2 = 81, so x = 9 or x = -9.
- For x^3 = 64: find a number that cubes to 64. 4^3 = 4 x 4 x 4 = 64, so x = 4 (the only real solution).
Answer: x^2 = 81 gives x = 9 or x = -9. x^3 = 64 gives x = 4.
- Why does x^2 = 81 have two solutions but x^3 = 64 has only one?
- What is the cube root of -8, and how do you know?
Common misconceptions and how to address them
Misconception0.999... is slightly less than 1, not actually equal to it.
Why it happens: Students trust the visual appearance of an infinite string of 9s over the algebra, assuming 'almost 1 forever' cannot equal 1 exactly.
How to address it: Apply the exact same repeating-decimal method used in this unit: let x = 0.999..., so 10x = 9.999..., and 10x - x = 9.999... - 0.999... = 9, giving 9x = 9, so x = 1. The method that turns 0.777... into 7/9 turns 0.999... into exactly 1, no approximation involved.
Misconceptiona^0 equals 0.
Why it happens: Students over-generalise from 'anything times 0 is 0' and expect an exponent of 0 to behave the same way.
How to address it: Use the quotient rule to show why: a^3 / a^3 = a^(3-3) = a^0, but a^3 / a^3 is also just 1 (anything divided by itself). So a^0 must equal 1, for any nonzero a.
MisconceptionA negative exponent makes the result negative.
Why it happens: Students associate the minus sign directly with the sign of the answer, rather than with 'flip to a fraction'.
How to address it: A negative exponent means reciprocal, not negative: a^-n = 1/a^n. Show 4^-2 = 1/16 side by side with -4^2 = -16 so the two very different meanings of a minus sign near an exponent do not get confused.
MisconceptionYou can combine 2^3 x 3^2 into a single power by adding the exponents, the same as 2^3 x 2^4.
Why it happens: Students apply the product rule mechanically without checking that the bases match first.
How to address it: The product and quotient rules only work when the BASE is the same on both powers. 2^3 x 3^2 = 8 x 9 = 72, which must be multiplied out directly; there is no shortcut when the bases differ.
MisconceptionCube roots of negative numbers do not exist, just like square roots of negative numbers.
Why it happens: Students over-generalise the rule that a negative number has no real square root to cube roots as well.
How to address it: A negative number cubed stays negative (since three negative factors multiply to a negative), so every negative number DOES have a real cube root. The cube root of -8 is -2, since (-2)^3 = -8; only even-power roots (square, fourth, ...) of negatives fail to exist in the reals.
Guided practice (with answers)
1. Is 5/8 rational or irrational?
Answer: Rational, because it is already written as one integer divided by another.
2. Convert 0.333... to an exact fraction.
Answer: 1/3, because if x = 0.333..., then 10x = 3.333..., and 10x - x = 3, so 9x = 3 and x = 3/9 = 1/3.
3. Simplify 3^4 x 3^2.
Answer: 729, because 3^4 x 3^2 = 3^(4+2) = 3^6 = 729.
4. Simplify 6^5 / 6^3.
Answer: 36, because 6^5 / 6^3 = 6^(5-3) = 6^2 = 36.
5. Evaluate 2^-3.
Answer: 1/8, because a negative exponent means the reciprocal: 2^-3 = 1/2^3 = 1/8.
6. Evaluate the cube root of 125.
Answer: 5, because 5^3 = 5 x 5 x 5 = 125.
Independent practice worksheets
Practise classifying numbers, exponent rules and roots with computed, never-wrong answer keys.
Differentiation
- Start the repeating-decimal conversion with a single-digit repeat (like 0.777...) before trying a two- or three-digit repeating block.
- Keep a reference card of the three exponent rules (product, quotient, power) with a worked example next to each while the pattern is new.
- Use only perfect squares and perfect cubes at first (1, 4, 9, 16, 25... and 1, 8, 27, 64...) before introducing negative-cube examples.
- Physically write out a small exponent expansion, like 2^3 x 2^4 = (2x2x2) x (2x2x2x2), so the 'add the exponents' rule is visibly true, not just memorised.
- Convert a repeating decimal with a non-repeating part first, such as 0.1666..., which needs an extra algebra step.
- Explore why sqrt(2) x sqrt(2) = 2 exactly, even though sqrt(2) itself has no exact decimal.
- Combine multiple exponent rules in one expression, such as (2^3 x 2^-1)^2 / 2^4.
- Investigate whether the sum or product of two irrational numbers is always irrational (it is not always; find a counterexample).
Assessment: exit ticket
A three-question exit ticket sampling classification, exponent rules, and roots.
1. Is sqrt(36) rational or irrational? Explain.
Answer: Rational, because sqrt(36) = 6 exactly (36 is a perfect square), and 6 can be written as 6/1.
2. Simplify 5^7 / 5^5.
Answer: 25, because 5^7 / 5^5 = 5^(7-5) = 5^2 = 25.
3. Evaluate the cube root of -27.
Answer: -3, because (-3)^3 = -3 x -3 x -3 = -27.
Teacher notes and timings
- Rough timing across four lessons: Lesson 1 rational vs irrational (section 1), Lesson 2 repeating decimals to fractions (section 2), Lesson 3 exponent rules (section 3), Lesson 4 roots plus the exit ticket (section 4 and assessment).
- This unit assumes comfort with square numbers and basic order of operations (Grade 7). Revisit those first if either is shaky.
- Language to repeat: rational means 'ratio-nal', a ratio of two integers; the exponent rules only combine powers that share the same base; a negative exponent flips to a fraction, it does not make the value negative.
- The 0.999... = 1 misconception is worth spending real time on: it is one of the most persistent misconceptions in mathematics at any level, and the repeating-decimal method taught in section 2 resolves it rigorously rather than by assertion.
- Curriculum note: 8.NS.A.1 (Common Core) covers classifying numbers and repeating decimals; 8.EE.A.1 covers the exponent laws; 8.EE.A.2 covers square and cube roots. All three are grouped here as the Grade 8 'number system foundations' students need before scientific notation (the next unit) and later algebra.
- Present and print both work: use the Print button for a clean handout, or work the repeating-decimal algebra live on the board, letting students suggest the multiplier for a new example.