Estimate and approximate square roots
Estimating and approximating square roots teaches pre-algebra students how to locate any square root between two consecutive perfect squares without a calculator. Because 8² = 64 and 9² = 81, students can determine that √70 falls between 8 and 9. For a closer approximation, a calculator rounds √17 to 4.12. This skill from OpenStax Prealgebra 2E bridges the gap between perfect squares and irrational numbers, helping students develop number sense around roots and preparing them for working with irrational numbers and the real number system.
Key Concepts
Property To estimate a square root, locate it between two consecutive perfect squares.
Examples To estimate $\sqrt{70}$, we know $8^2 = 64$ and $9^2 = 81$. Since 70 is between 64 and 81, we know that $8 < \sqrt{70} < 9$. Using a calculator to find $\sqrt{17}$ and rounding to two decimal places gives the approximation $\sqrt{17} \approx 4.12$. To estimate $\sqrt{172}$, notice it is between the perfect squares $169$ ($13^2$) and $196$ ($14^2$). Therefore, $13 < \sqrt{172} < 14$.
Explanation Not all numbers have whole number square roots. For these non perfect squares, we can estimate the root's value by identifying the two closest perfect squares. A calculator provides a more precise but still approximate decimal value.
Common Questions
How do you estimate a square root without a calculator?
Identify the two perfect squares closest to your number. Since 8² = 64 and 9² = 81, the square root of 70 falls between 8 and 9.
What is the approximate value of √17?
Using a calculator and rounding to two decimal places, √17 ≈ 4.12.
Why can't most square roots be expressed exactly?
Numbers that are not perfect squares have square roots that are irrational — their decimal form never terminates or repeats.
How do you estimate √172?
Since 13² = 169 and 14² = 196, √172 falls between 13 and 14, closer to 13.
What are perfect squares and why do they matter here?
Perfect squares (1, 4, 9, 16, 25, …) are numbers with whole-number square roots. They serve as benchmarks for estimating all other square roots.
What course covers estimating square roots in OpenStax?
This skill appears in OpenStax Prealgebra 2E, typically studied in 7th or 8th grade pre-algebra courses.