Square & Cube Roots Reference Cheat Sheet

Key Facts

• A square root asks what number was squared, so $\sqrt{49}=7$ because $7^2=49$.
• A cube root asks what number was cubed, so $\sqrt[3]{64}=4$ because $4^3=64$.
• Perfect squares include $1,4,9,16,25,36,49,64,81,100,121,144$.
• Perfect cubes include $1,8,27,64,125,216,343,512,729,1000$.
• For nonnegative numbers, squaring and square roots undo each other: $\sqrt{a^2}=a$ when $a\ge 0$.
• Cube roots can be negative because $\sqrt[3]{-27}=-3$ and $(-3)^3=-27$.
• To estimate $\sqrt{n}$, find perfect squares around it, such as $36<40<49$, so $6<\sqrt{40}<7$.
• To estimate $\sqrt[3]{n}$, find perfect cubes around it, such as $125<150<216$, so $5<\sqrt[3]{150}<6$.

Vocabulary

Square root

A square root of a number is a value that gives the original number when multiplied by itself.

Cube root

A cube root of a number is a value that gives the original number when multiplied by itself three times.

Radical

A radical is the symbol used to show a root, such as $\sqrt{x}$ or $\sqrt[3]{x}$.

Radicand

The radicand is the number or expression inside a radical symbol.

Perfect square

A perfect square is a number that can be written as $n^2$ for a whole number $n$.

Perfect cube

A perfect cube is a number that can be written as $n^3$ for a whole number $n$.