Pre-Algebra
Numbers and operations
Square roots
20 practice questions
0 video lessons
Theory + worked examples
Square Roots
California Pre-Algebra • Standard 8.EE.2 • Numbers & Operations
Square Roots is a topic in Numbers & Operations in the California Common Core State Standards. It is aligned to Standard 8.EE.2, which requires students to evaluate square roots of perfect squares and estimate others.
A square root \(\sqrt{a}\) is the non-negative number whose square is \(a\); perfect squares have whole-number roots.
Theory
The square root \(\sqrt{a}\) is the non-negative number whose square is \(a\).
Perfect squares like \(1,4,9,16,25\) have whole-number roots; others fall between integers.
What a square root is.
Perfect squares as areas.
Definition:
\[\sqrt{a}=b \iff b^2=a\ \ (b\ge0)\]
Estimate non-perfect squares between nearby roots.
How to find a square root
- Ask what number squared gives the value.
- Recognize perfect squares.
- For others, bracket between perfect squares.
- Root numerator and denominator of a fraction separately.
Example 1 — Perfect square
Find \(\sqrt{49}\).
Solution
\(7^2=49\).
| \(\sqrt{49}\) | \(=\) | \(7\) |
Example 2 — Two roots idea
What number squared gives \(64\)?
Solution
\(8^2=64\), so the principal square root is \(8\).
| \(\sqrt{64}\) | \(=\) | \(8\) |
Example 3 — Estimate
Between which integers is \(\sqrt{20}\)?
Solution
\(16<20<25\), so between \(4\) and \(5\).
| \(4\) | < | \(\sqrt{20}<5\) |
Example 4 — Fraction
Find \(\sqrt{\dfrac{9}{16}}\).
Solution
Root the top and bottom.
| \(\dfrac{\sqrt9}{\sqrt{16}}\) | \(=\) | \(\dfrac{3}{4}\) |
Common pitfalls
\(\sqrt{20}\) is not \(10\) — it is about \(4.47\).
The principal root is the non-negative one.
\(\sqrt{a+b}\neq\sqrt a+\sqrt b\).
Frequently asked questions
What is a square root?
A number that, squared, gives the original.
What is a perfect square?
A number whose square root is a whole number.
What is \(\sqrt{49}\)?
\(7\).
Is \(\sqrt{20}\) a whole number?
No, it is between \(4\) and \(5\).
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