Lesson 2: Square Roots and Cube Roots

New Concept

This lesson introduces square and cube roots, the inverse of squaring and cubing. We'll learn radical notation ($\sqrt{b}$, $\sqrt[3]{b}$), differentiate rational vs. irrational numbers, and use the order of operations for radical expressions.

What’s next

This card is just the intro. Up next, you'll tackle practice cards and interactive examples to master calculating roots and simplifying radical expressions.

Property

The number $s$ is called a square root of a number $b$ if $s^2 = b$. Every positive number has two square roots, one positive and one negative.

Examples

• The two square roots of 49 are 7 and -7, because $7^2 = 49$ and $(-7)^2 = 49$.
• A square garden has an area of 64 square feet. Its side length is a square root of 64, which is 8 feet.
• Since $12^2 = 144$, we know that 12 is a square root of 144. The other square root is $-12$.

Explanation

Finding a square root is the reverse of squaring a number, like finding a square's side from its area.

Property

The positive square root of a number is called the principal square root. The symbol $\sqrt{\ }$ denotes the positive or principal square root. The symbol $\sqrt{\ }$ is called a radical sign, and the number inside is called the radicand.

Examples

• The principal square root of 81 is written as $\sqrt{81}$, which equals 9.
• To express the negative square root of 36, we write $-\sqrt{36}$, which equals $-6$.
• The expression $\pm\sqrt{100}$ represents both square roots of 100, which means 10 or $-10$.

Explanation

The radical symbol $\sqrt{\ }$ is a specific instruction to find only the positive square root, known as the principal root.

Property

A rational number is one that can be expressed as a quotient (or ratio) of two integers, where the denominator is not zero. An irrational number is one that cannot be expressed as a quotient of two integers. The decimal representation of an irrational number never ends, and does not repeat any pattern.

Examples

• The number $0.75$ is rational because it can be written as the fraction $\frac{3}{4}$.
• The number $\frac{5}{11}$ is rational. Its decimal form, $0.\overline{45}$, repeats the digits 4 and 5 forever.
• The value of $\sqrt{7}$ is an irrational number. Its decimal approximation starts $2.6457...$ and continues infinitely without a repeating block of digits.

Explanation

Rational numbers are numbers that can be written as a simple fraction. Their decimal form either terminates or repeats.

Property

1. Perform any operations inside parentheses, or above or below a fraction bar.
2. Compute all indicated powers and roots.
3. Perform all multiplications and divisions in the order in which they occur from left to right.
4. Perform additions and subtractions in order from left to right.

Examples

• To simplify $10 + 4\sqrt{25}$, first find the root: $\sqrt{25}=5$. Then multiply: $4 \times 5 = 20$. Finally, add: $10+20=30$.
• To evaluate $\sqrt{9^2 - 8(5)}$, first calculate inside the radical: $\sqrt{81 - 40} = \sqrt{41}$. The exact answer is $\sqrt{41}$.
• In the expression $\frac{20 - \sqrt{16}}{2}$, first evaluate the root $\sqrt{16}=4$. Then subtract in the numerator: $20-4=16$. Finally, divide: $\frac{16}{2}=8$.

Explanation

In the order of operations, roots share the same level of priority as exponents.

Property

The number $c$ is called a cube root of a number $b$ if $c^3 = b$. Every number has exactly one cube root. The cube root of a positive number is positive, and the cube root of a negative number is negative.

Examples

• The cube root of 27 is 3, written as $\sqrt[3]{27} = 3$, because $3^3 = 27$.
• The cube root of $-64$ is $-4$, written as $\sqrt[3]{-64} = -4$, because $(-4)^3 = -64$.
• A cube-shaped box has a volume of 1000 cubic inches. Its side length is the cube root of the volume, so $\sqrt[3]{1000} = 10$ inches.

Explanation

Finding a cube root is the reverse of cubing a number (raising it to the third power).