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.

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 \(\boldsymbol{4}\) 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.

---