Roots (square, cube, etc.)

Just like we have exponents and raise a number to the power of something, we can also take the square root of a number.

Square roots are denoted by the symbol ${\displaystyle {\sqrt {}}}$ , which is called a radical—there is a number present underneath the radical and that is the number that we are "taking the square root" of.

Square roots

By taking the "square root" of a number, we are essentially undoing an exponent that has the power of 2. Square roots and squaring a number are the exact opposite of each other and come in handy when solving equations.

Just like we have ${\displaystyle 2^{2}=4}$ where we raise the base number of 2 to the power of 2, we can take the square root of the answer of 4 to obtain two: ${\displaystyle {\sqrt {4}}=2}$. By definition, the same number multiplied by itself is a square and by taking the square root, we can find that square.

Of our first 12 perfect squares, we can take the square roots of the answer to "undo" the square.

• ${\displaystyle {\sqrt {1}}=1}$
• ${\displaystyle {\sqrt {4}}=2}$
• ${\displaystyle {\sqrt {9}}=3}$
• ${\displaystyle {\sqrt {16}}=4}$
• ${\displaystyle {\sqrt {25}}=5}$
• ${\displaystyle {\sqrt {36}}=6}$
• ${\displaystyle {\sqrt {49}}=7}$
• ${\displaystyle {\sqrt {64}}=8}$
• ${\displaystyle {\sqrt {81}}=9}$
• ${\displaystyle {\sqrt {100}}=10}$
• ${\displaystyle {\sqrt {121}}=11}$
• ${\displaystyle {\sqrt {144}}=12}$

It is worth noting that the answer of a square root will cause you to obtain both the positive and negative versions of the integer. For example, we know that ${\displaystyle {\sqrt {4}}=2}$ because ${\displaystyle 2*2=4}$, but also ${\displaystyle {\sqrt {4}}=-2}$ because ${\displaystyle -2*(-2)=4}$ because a negative multiplied by a negative will equal a positive. So when we write the answer of a square root, oftentimes we write ${\displaystyle \pm }$ in front of our integer unless asked otherwise.

Cube roots

If square roots are the way to undo squaring a number, then cube roots are the way to undo cubing a number.

Remember that cubing a number means raising it to the power of 3—or multiplying it by itself 3 times. So when we take the cubic root of a number, we want to find out what number multiplied by itself three times will equal our number under the radical.

For example, ${\displaystyle {\sqrt[{3}]{8}}=2}$. When writing a cubic root, to signify that it is a cubic root, we must write the 3 on the outside left corner of the radical. We do not do this with square roots, as it is implied that the presence of a radical means square root. Back to the example, we know that the answer must be one that when multiplied by itself three times, it will equal 8. Since ${\displaystyle 2*2*2=8}$, we can come to the conclusion that the cubic root of 8 is 2.

Larger roots

It is unusual for there to be questions about larger roots by hand. Most of the larger roots are handled by a calculator. The larger roots follow a similar premise as both square roots and cubic roots though: we must find an integer that when multiplied by itself 4 times, or however many times our root is, equals the number underneath the radical.

For more: Return to Pre-Algebra