Number & operations · Grade 8
Square Roots and Irrational Numbers
Square roots reverse squaring. Some square roots are rational numbers, while others cannot be written as a fraction of integers.
Learning time: 20–25 minutes · Level: Developing
Learning goals
- Identify the mathematical relationship or rule.
- Apply it accurately in worked and new problems.
- Check calculations and explain your reasoning.
1. Perfect squares
A perfect square is the square of an integer: 1, 4, 9, 16, 25, 36, and so on. The square root of 49 is 7 because 7² = 49.
2. Estimate square roots
If a number lies between two perfect squares, its square root lies between their roots. Since 25 < 30 < 36, √30 is between 5 and 6.
3. Rational numbers
A rational number can be written as a/b where a and b are integers and b ≠ 0. Integers, fractions and terminating or repeating decimals are rational.
4. Irrational numbers
An irrational number cannot be written as a ratio of integers. Its decimal expansion continues without terminating or repeating. Examples include √2 and π.
5. Compare real numbers
Use nearby perfect squares or decimal estimates to compare roots. For example, √50 is greater than 7 because 50 > 49, but less than 8 because 50 < 64.
Estimate and compare square roots
Use nearby perfect squares to estimate a square root. If 36 < 48 < 49, then 6 < √48 < 7. Since 48 is much closer to 49 than to 36, √48 is close to 7. A calculator gives an approximate decimal, but the exact value √48 can be simplified to 4√3.
Worked reasoning: To compare √48 and 6, square the positive numbers: 6² = 36. Since 48 is greater than 36, √48 is greater than 6. This comparison avoids estimating a decimal.
A square root of a perfect square is an integer, so √81 = 9 is rational. The square root of a positive integer that is not a perfect square, such as √11, is irrational. Its decimal continues without ending or repeating.
- Do not assume every square root is irrational.
- When comparing positive square roots, compare the numbers inside the roots.
- Use an approximation sign (≈), not an equals sign, for rounded decimals.
Worked examples
- √81 = 9 because 9² = 81.
- √20 lies between 4 and 5 because 16 < 20 < 25.
- √2 ≈ 1.414… and is irrational.
- 0.375 = 3/8, so it is rational.
Independent practice
Attempt each question before reviewing the answer.
- Find √121. Answer: 11.
- Between which integers is √40? Answer: Between 6 and 7.
- Is 0.272727… rational? Answer: Yes; it repeats.
- Is √11 rational or irrational? Answer: Irrational.
- Which is greater, √48 or 6? Answer: √48 is greater than 6 because 48 is greater than 36.
- Write 0.6 as a fraction. Answer: 3/5.
- Is 7 an integer and rational? Answer: Yes; 7 = 7/1.
- Between which integers is √70? Answer: Between 8 and 9.
Knowledge check
Choose one answer for each question, then check for explanations.
Printable worksheet and answer key
Use the independent-practice questions above as your worksheet. Show each step and check your answers after completing the set.
Study reminder
Write the rule used beside each solution to make your reasoning clear.