Roots, powers and exponents

Square numbers

A square number is a number multiplied by itself. The symbol is ² (we call it exponent). The number squared is called the base:

We can read this as two/three/four to the second power.

A cube number is a number multiplied by itself 3 times:

Square Roots

The square root of a number is a number which, when multiplied by itself, gives the original number. For example:

Therefore:

When we take the square root of a number, we use only the non-negative root. Therefore:

However, when we look for numbers whose square is 16, both a positive and a negative number work:

Therefore, the equation

has two solutions:

So it is important to distinguish between taking a square root and solving an equation with a square:
$√16 = 4$, but $x^2 = 16$ has the solutions $x = 4$ and $x = -4$.

The number inside the square root symbol $√$ is called the radicand. For example, in $√64$, 64 is the radicand.

For higher roots, we write the index above the radical sign. For example, in $√^3{64};$, 64 is the radicand and 3 is the index.

Exponents and Powers

An exponent tells us how many times a number is multiplied by itself. The number being multiplied is called the base, and the small raised number is called the exponent. The whole expression is called a power.

For example, in $2^4$, the number 2 is the base and the number 4 is the exponent. The expression $2^4$ is a power and is read as “two to the fourth power”.

The exponent 4 tells us to use the base 2 as a factor four times. So we can say that two to the fourth power is sixteen.

Here, the exponent 3 tells us to multiply 3 by itself three times. We read $3^3$ as “three to the third power”. We can also say “three cubed”.

The exponent 2 also has a special name. For example, $5^2$ can be read as “five to the second power” or “five squared”.

Special exponents

There are several important exponent rules that we need to remember.

Any non-zero number raised to the exponent 0 equals 1:

For example:

A number raised to the exponent 1 stays the same:

For example:

Negative exponents

A negative exponent means that we take the reciprocal of the corresponding positive power.

For example:

This rule applies when the base is not zero.

Fractional exponents

A fractional exponent represents a root. An exponent of $1/n$ means the n-th root of the number.

For example, an exponent of $1/2$ means a square root:

An exponent of $1/3$ means a cube root:

Simplifying square roots

To simplify roots you can use Product Property of Square Roots:

At first, find the perfect square inside the square root:

This can be written as $2√6$.

Simplifying square roots

To simplify roots you can use Product Property of Square Roots:

At first, find the perfect square inside the square root:

This can be written as $2√6$.

You can also use these formulae: