1.08 Square roots and perfect squares

We have previously learned that a perfect square is a whole number that can be created by multiplying an integer by itself. Now we will explore square roots.

If we are asked to find the square root of a value, we are being asked, "What positive number multiplied by itself would give this value?" It is also the same as asking "What positive number, when squared, would give this value?"

The notation we use is the square root symbol, \sqrt{}, with a number inside it. For example, the square root of 25 is written as \sqrt{25} and it is equivalent to 5 because 5 \cdot 5 =25.

We can visualize the square root of a number as the side length of a square whose area is equal to the number.

For 25 we can set up a square with an area of 25 square units.

Counting the side lengths, we can find that each side is made up of 5 squares. This shows \sqrt{25}=5.

Examples

Example 1

Evaluate \sqrt{256}.

Worked Solution

Create a strategy

We need to find a positive number that equals 256 when multiplied by itself. We can start from a perfect square that we have memorized, such as 12^2=144, then square numbers larger than 12.

Apply the idea

256 is larger than 144, which is 12^2, so we can begin by squaring 13,\,14,\,15,\, etc. until we find the solution.

\displaystyle 13\cdot13	\displaystyle =	\displaystyle 169
\displaystyle 14\cdot14	\displaystyle =	\displaystyle 196
\displaystyle 15\cdot 15	\displaystyle =	\displaystyle 225
\displaystyle 16 \cdot 16	\displaystyle =	\displaystyle 256

The answer is 16.

Reflect and check

We can also use a grid to verify our answer. If we create a square grid with 256 smaller squares, we should be able to see that each side length is 16.

Watch question walkthrough