Lesson 7: Simplify and Use Square Roots

New Concept

This lesson introduces the square root, the inverse of squaring a number. You'll learn how to find values like $\sqrt{25}$, estimate others like $\sqrt{60}$, and apply this skill to solve real-world problems.

What’s next

Now that you have the basics, you'll tackle interactive examples, practice problems, and real-world challenges to master using square roots.

Property

If $n^2 = m$, then $m$ is the square of $n$. A perfect square is the square of a whole number. Squaring a positive or negative number results in a positive number.

Examples

• The square of 8 is 64, because $8^2 = 8 \times 8 = 64$.
• The number 144 is a perfect square, as it is the square of a whole number: $12^2 = 144$.
• The square of $-5$ is 25, since $(-5)^2 = (-5)(-5) = 25$.

Explanation

Squaring a number means multiplying it by itself. This concept is named after a geometric square, where the area is the side length multiplied by itself. Perfect squares are the result of squaring whole numbers, like 9 from $3^2$.

Property

A number whose square is $m$ is called a square root of $m$. If $n^2 = m$, then $n$ is a square root of $m$. The notation $\sqrt{m}$ is read as 'the square root of $m$' and represents the positive square root, or principal square root. For $n \ge 0$, if $m = n^2$, then $\sqrt{m} = n$.

Examples

• Since $5^2 = 25$, the principal square root of 25 is $\sqrt{25} = 5$.
• To indicate the negative square root of 100, we write $-\sqrt{100}$, which simplifies to $-10$.
• Zero has only one square root. Because $0^2 = 0$, we have $\sqrt{0} = 0$.

Explanation

Finding a square root is the inverse operation of squaring a number. The radical symbol $\sqrt{m}$ asks for the positive number that, when multiplied by itself, equals $m$. Every positive number actually has two square roots: one positive and one negative.

Property

When using the order of operations, treat the radical sign as a grouping symbol. Simplify any expressions under the radical sign before performing other operations. For any negative number, there is no real number solution for its square root.

Examples

• To simplify $\sqrt{9 + 16}$, you must first add the numbers inside the radical to get $\sqrt{25}$, which equals $5$.
• To simplify $\sqrt{9} + \sqrt{16}$, you find each square root separately before adding: $3 + 4 = 7$. Note this is different from the first example.
• The expression $\sqrt{-169}$ is not a real number because no real number multiplied by itself can result in $-169$.

Explanation

The radical sign acts like parentheses, so you must simplify the expression inside it first. You cannot take the square root of a negative number in the real number system because squaring any real number always results in a non-negative value.

Property

To estimate a square root, locate it between two consecutive perfect squares.

Examples

• To estimate $\sqrt{70}$, we know $8^2 = 64$ and $9^2 = 81$. Since 70 is between 64 and 81, we know that $8 < \sqrt{70} < 9$.
• Using a calculator to find $\sqrt{17}$ and rounding to two decimal places gives the approximation $\sqrt{17} \approx 4.12$.
• To estimate $\sqrt{172}$, notice it is between the perfect squares $169$ ($13^2$) and $196$ ($14^2$). Therefore, $13 < \sqrt{172} < 14$.

Explanation

Not all numbers have whole number square roots. For these non-perfect squares, we can estimate the root's value by identifying the two closest perfect squares. A calculator provides a more precise but still approximate decimal value.

Property

To simplify a square root containing a variable, find an expression that, when squared, equals the expression inside the radical. In this context, we assume all variables in a square root expression are non-negative.

Examples

• To simplify $\sqrt{y^2}$, we recognize that $(y)^2 = y^2$, so $\sqrt{y^2} = y$.
• To simplify $\sqrt{49x^2}$, find the square root of 49 (which is 7) and the square root of $x^2$ (which is $x$), giving a result of $7x$.
• To simplify $-\sqrt{100y^2}$, the negative sign stays outside. Since $\sqrt{100y^2} = 10y$, the expression simplifies to $-10y$.

Explanation

When simplifying square roots with variables, think 'what expression, when squared, gives me this?'. Assuming variables are non-negative, $\sqrt{x^2}$ simplifies to $x$. You can find the square root of the coefficient and the variable part separately.

Property

Square roots are used in various real-world formulas.

Examples

• If a square patio has an area of 200 square feet, the length of one side is $\sqrt{200} \approx 14.1$ feet.
• If a pair of sunglasses is dropped from a bridge 400 feet high, the time it takes to reach the river is $\sqrt{\frac{400}{4}} = \sqrt{100} = 5$ seconds.
• If a car's skid marks measure 190 feet, its speed before braking was $\sqrt{24 \cdot 190} = \sqrt{4560} \approx 67.5$ miles per hour.

Explanation

Square roots are essential for solving practical problems, especially when you need to reverse a squaring operation. This is common in geometry for finding lengths from areas and in physics for formulas involving time, speed, and distance.