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Evaluation of Square Roots

Find the value of each square root.

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Evaluation of Square Roots
License: CC BY-NC 3.0

Brittany is replacing the old tiles on a small tabletop for her mom's upcoming birthday. The area that needs to be covered is 64 square inches, so Brittany bought new 1-inch square tiles. How many rows of tiles will she need to cover the table? How many tiles will be in each row?

In this concept, you will learn what a square root is and how to find it.

Evaluating Square Roots

A square is a unique shape in that all four sides are equal in length.

In order to find the area of the square (the number of units it covers), you can "square" one side.

\begin{align*}A={s}^{2}\end{align*} 

To "square" a number means to multiply it by itself.

This type of calculation requires multiplying the parts (side lengths) in order to find the whole (area). 

Let's look at an example.

If you have a square with 6" sides, you multiply 6" x 6", or \begin{align*}{6}^{2}\end{align*}, to get an area of 36 square inches (Don't forget that when you multiply the units of inches, they also get squared. Your answer is not inches, but rather inches squared.) 

This is the symbol used to find the square root of x.  \begin{align*}\sqrt{x}\end{align*}

A square root is a part that when you multiply it by itself (the other part) equals the whole.The root of a square is one of its parts (the length of one side). Finding the "root" of a square is the opposite calculation of squaring. In order to find it, you will be given the whole (the area), and you will need to find the parts. The good news is that the parts in a square are always equal!

All numbers have a square root, even those that are not perfect squares. A perfect square is a number for which the roots are whole numbers. These are found in your multiplication tables.

All positive numbers also have a negative square root. Remember that when you multiply a negative number times a negative number, the answer is a positive. Positive numbers will have both a  positive and negative root of the same absolute value.

Let's look at some more examples.

\begin{align*}\sqrt{16}=4\end{align*} 

The root, 4, is a whole number. Therefore, the number 16 is a perfect square.

 \begin{align*}\sqrt{25}=5\end{align*} 

The number 25 is a perfect square.

 \begin{align*}\sqrt{20}=4.47\end{align*}

This is a decimal root. The number 20 is NOT a perfect square. Its root is somewhere between the whole numbers of 4 and 5.                              

Examples

Example 1

Earlier, you were given a problem about Brittany and the tiles for her table.

She has 1" square tiles and needs to cover the 64 square inch table top. How many tiles will be in each row, and how many rows does she need?

First, you know that the perfect square \begin{align*}64=8\times 8\end{align*} , or  \begin{align*}{8}^{2}\end{align*}.

The answer is \begin{align*}\sqrt{64}=8\end{align*}. Brittany should have 8 rows of tiles with 8 tiles in each row.

Example 2

Find the square root of 9.

First, remember that you are working with a perfect square; all sides are equal.

Next, picture that square of 9 units in your mind.

Then think, "What number do you multiply by itself (or "square") to get 9?

The answer is 3.

The square root of 9, \begin{align*}\sqrt{9}\end{align*}, is 3 because \begin{align*}3^2\end{align*}, or \begin{align*}3 \times 3\end{align*}, is equal to 9. 

Example 3

Find  \begin{align*}\sqrt{49}\end{align*}

First, as the numbers get bigger, it may be difficult to picture them in your mind, so think of the perfect squares that appear in your multiplication tables.

Next, ask yourself, "what number times itself equals 49?"

 \begin{align*}5\times 5=25\end{align*}

 \begin{align*}6\times 6=36\end{align*}

Then, \begin{align*}7\times 7=49\end{align*}

The answer is 7.

 \begin{align*}\sqrt{49}=7\end{align*}

Example 4

Evaluate \begin{align*}\sqrt{21}\end{align*}

First, the whole number 21 does not appear in your multiplication tables because it is not a perfect square.

Next, think about the perfect squares it may fall between.

 \begin{align*}{4}^{2}=16\end{align*}

 \begin{align*}{5}^{2}=25\end{align*}

21 is between the perfect squares of 16 and 25.

The answer is  \begin{align*}\sqrt{21}\end{align*}  is between 4 and 5. 

Example 5

The YMCA is collecting artwork by children to hang on the wall of their art center. Each submission must cover a square space of 100 square inches. What size sheet of paper should Brent use to paint a self portrait?

First, think about what information is given to you. The drawing must be square, and it must cover 100 square inches. To know the length of the sides, you must find the square root of 100.

Then, think of the perfect squares in your multiplication tables.

 \begin{align*}{9}^{2}=81\end{align*}

 \begin{align*}{10}^{2}=100\end{align*}

The answer is \begin{align*}\sqrt{100}=10\end{align*}.  Brent needs a sheet of paper that measures 10 inches by 10 inches.

Review

Evaluate each square root.

  1. \begin{align*}\sqrt{16}\end{align*}
  2. \begin{align*}\sqrt{25}\end{align*}
  3. \begin{align*}\sqrt{1}\end{align*}
  4. \begin{align*}\sqrt{49}\end{align*}
  5. \begin{align*}\sqrt{144}\end{align*}
  6. \begin{align*}\sqrt{81}\end{align*}
  7. \begin{align*}\sqrt{169}\end{align*}
  8. \begin{align*}\sqrt{121}\end{align*}
  9. \begin{align*}\sqrt{100}\end{align*}
  10. \begin{align*}\sqrt{256}\end{align*}

Name the two values each square root is between.

  1. \begin{align*}\sqrt{12}\end{align*}
  2. \begin{align*}\sqrt{14}\end{align*}
  3. \begin{align*}\sqrt{30}\end{align*}
  4. \begin{align*}\sqrt{40}\end{align*}
  5. \begin{align*}\sqrt{50}\end{align*}
  6. \begin{align*}\sqrt{62}\end{align*}
  7. \begin{align*}\sqrt{70}\end{align*}
  8. \begin{align*}\sqrt{101}\end{align*}
  9. \begin{align*}\sqrt{5}\end{align*}\begin{align*}\sqrt{15}\end{align*}

Review (Answers)

To see the Review answers, open this PDF file and look for section 9.1.

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Vocabulary

Congruent

Congruent figures are identical in size, shape and measure.

Square

A square is a polygon with four congruent sides and four right angles.

Square Number

A square number or perfect square is a whole number that is the square of another integer. For example, 36 is a square number because 6 \cdot 6 = 36.

Square Root

The square root of a term is a value that must be multiplied by itself to equal the specified term. The square root of 9 is 3, since 3 * 3 = 9.

Image Attributions

  1. [1]^ License: CC BY-NC 3.0

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