Lesson 5: Solve Equations Using Square Roots and Cube Roots

Property

Taking a square root is the opposite of squaring a number.
To solve an equation of the form $x^2 = k$ (where $k > 0$), we take the square root of both sides.
Because a positive number has two square roots, the solution is written as:

Examples

• To solve the equation $x^2 = 81$, we take the square root of both sides. The solutions are $x = \pm\sqrt{81}$, which means $x = 9$ and $x = -9$.

Property

When solving quadratic equations using square roots, the nature of the solution depends on whether the radicand is a perfect square:

• Perfect square radicand: $x = \pm\sqrt{n^2} = \pm n$ (integer solutions)
• Non-perfect square radicand: $x = \pm\sqrt{a}$ where $a$ is not a perfect square (irrational solutions)

Examples

Property

$b$ is the cube root of $a$ if $b$ cubed equals $a$. In symbols, we write

Unlike square roots, which are not real for negative numbers, every real number has a real cube root. Simplifying radicals occurs at the same level as powers in the order of operations.

Examples

• To simplify $3\sqrt[3]{-8}$, we find the cube root of $-8$ which is $-2$, and then multiply by $3$. So, $3\sqrt[3]{-8} = 3(-2) = -6$.
• To evaluate $2 - \sqrt[3]{-125}$, we first find that the cube root of $-125$ is $-5$. The expression becomes $2 - (-5) = 7$.
• To simplify $\frac{10 + \sqrt[3]{-27}}{7}$, we calculate $\sqrt[3]{-27} = -3$. The expression becomes $\frac{10 + (-3)}{7} = \frac{7}{7} = 1$.

Explanation

A cube root is the inverse operation of cubing a number. Think of it as asking: 'What number, when multiplied by itself three times, gives me this value?' Unlike square roots, you can take the cube root of negative numbers.