## Lesson 5: Exploring Problems with Perfect Squares and Square Roots

Practice algebra with perfect squares and square roots:

$$\textcolor{blue}{\sqrt{25}} + \textcolor{red}{3^2}$$

$$=\textcolor{blue}{5} + \textcolor{red}{9}$$

$$= 14$$

Why? Remember, square roots and exponents are mathematical notations. Once you evaluate the expression it is easy to add them together.

Adding $$5$$ to the square of a number is $$30$$. What is the number?

Let $$x$$ represent the number:

$$\textcolor{blue}{5} + \textcolor{red}{x^2} = \textcolor{purple}{30}$$

$$\cancel{\textcolor{blue}{5}} - \cancel{\textcolor{blue}{5}} + \textcolor{red}{x^2} = \textcolor{purple}{30} - \textcolor{blue}{5}$$

$$\textcolor{red}{x^2} = \textcolor{orange}{25}$$

$$\textcolor{red}{x} = \textcolor{orange}{\sqrt{25}}$$

$$\textcolor{red}{x} = \textcolor{orange}{5}$$

Why? You can represents "the square of a number" by writing $$x^2$$.

Rank the following from largest to smallest:

$$\textcolor{red}{7}, \textcolor{blue}{3^2}, \textcolor{brown}{\sqrt{25}}, \textcolor{green}{\sqrt{64}}, \textcolor{purple}{\sqrt{9} + 1}, \textcolor{orange}{2 \cdot 1^2}$$

First, evaluate each number:

$$\textcolor{red}{7}, \textcolor{blue}{9}, \textcolor{brown}{5}, \textcolor{green}{8}, \textcolor{purple}{4}, \textcolor{orange}{2}$$

Now, rank the numbers from largest to smallest:

$$\textcolor{blue}{9}, \textcolor{green}{8}, \textcolor{red}{7}, \textcolor{brown}{5}, \textcolor{purple}{4}, \textcolor{orange}{2}$$

Now, express using the numbers using the original expressions:

$$\textcolor{blue}{3^2}, \textcolor{green}{\sqrt{64}}, \textcolor{red}{7}, \textcolor{brown}{\sqrt{25}}, \textcolor{purple}{\sqrt{9} + 1}, \textcolor{orange}{2 \cdot 1^2}$$

Why? Again, by evaluating the square roots and exponents it becomes much easier to compare the numbers.

#### Review these lessons:

The area of a square tablet screen is 144. What is the length?

Let $$l$$ represent the length of the tablet:

 $$l$$ $$l$$ $$A = l^2$$

$$144 = l^2$$

$$l = \sqrt{144}$$

$$l = 12$$

Why? A square has equal sides. The area of a square is $$A = l^2$$.

Suppose you have 20 apples and arrange them in a square and have a few left over. How many rows are in the square?

Now arrange them in a square. A few will be left over. Start with a 2 by 2 square.

There are not enough apples to make a larger square.

There are 4 rows and 4 columns in the square that contains 16 of the 20 apples. There are 4 apples left over.

Why? To make a square, use square numbers! $$2^2 = 4, 3^2 = 9, 4^2 = 16, 5^2 = 25$$. 16 is the largest square number that can be made with 20 apples without going over.