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.

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?

Start with 20 apples:

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.