Key takeaway: The square root of a number is a value that, multiplied by itself, gives that number, and the radical sign \(\sqrt{\ }\) denotes the nonnegative one.
Grade 88.EE.A.2Numbers and Operations
Definition of Square Root
The square root undoes squaring. Since \(12 \times 12 = 144\), the square root of 144 is 12. Geometrically, if a square has area 144, each side is 12 units long, which is why the operation is named after the shape.
Every positive number has two square roots, 12 and \(-12\) in this case, because a negative times a negative is positive. The radical symbol \(\sqrt{\ }\) asks only for the nonnegative one, called the principal root. When the number is not a perfect square, the root is irrational, so you estimate it between two perfect squares and refine with decimals.
Formula and notation
The radical sign gives the principal square root, and the equation \(x^2 = a\) gives both:
\[\sqrt{a} = b \ \text{ means } \ b^2 = a,\ b \ge 0\]
To solve \(x^2 = 49\), write \(x = \pm 7\).
Key facts
- The radical symbol returns only the nonnegative root; \(\sqrt{49} = 7\), not \(\pm 7\).
- Roots of perfect squares like 25, 64 and 121 are whole numbers.
- Other roots are irrational and are estimated between neighboring perfect squares.
- A real square root of a negative number does not exist.
Square Root example
A square garden bed in a community plot has an area of 175 square feet. Estimate the length of one side to the nearest tenth of a foot.
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The side is \(\sqrt{175}\). Since \(13^2 = 169\) and \(14^2 = 196\), the side is between 13 and 14.
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Test 13.2: \(13.2^2 = 174.24\). Test 13.3: \(13.3^2 = 176.89\).
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175 is much closer to 174.24 than to 176.89, so the side is about 13.2.
Each side is about 13.2 feet long.
Common mistake
Watch out
Zane simplifies \(\sqrt{9 + 16}\) as \(\sqrt{9} + \sqrt{16} = 3 + 4 = 7\).
Why it is wrong
A square root does not split across addition. First add under the radical: \(\sqrt{25} = 5\). Check by squaring his answer: \(7^2 = 49\), not 25. Splitting works for multiplication, such as \(\sqrt{9 \cdot 16} = 3 \cdot 4\), but never for sums.
Related terms and lessons
Browse the glossary: A–Z list · Grade 8 vocabulary · Numbers and Operations