Part 1: Perfect Squares and Square Roots_L08_U02_C3 Math 2023_Ready for Review

Connections 3 Lesson 8: Square and Cube Roots Part 1: Perfect Squares and Square Roots_L08_U02_C3 Math 2023_Ready for Review

Part 1: Perfect Squares and Square Roots
Complete Part 1 of your Lesson 8 Interactive Notes.

Perfect squares are the products of a number multiplied by itself. They are called perfect squares because when counted out in units, they form a perfect square. Numbers that are squared have an exponent of 2 or are raised to a power of 2.

For example, 9 is a perfect square. Let’s take a look.

Perfect Squares

Example 1
The product of $3 \cdot 3$ or $3^{2}$ is 9. $3 \times 3$ When 3 is multiplied by itself it is squared.

Can you create a perfect square with 9 units?

Try it out! Arrange nine Math U See Manipulative Blocks to form a square.
Record your model in Part 1 of your Lesson 8 Interactive Notes.

When you are ready, click the solution button below to compare your solution.

Solution

**The content below should appear when the solution button is clicked.

A perfect square can be created using 9 units.

As you can see, there are 3 units aligned horizontally across the top of the array, with another 3 units aligned top to bottom vertically along the array. This means that 9 is a perfect square of 3.

$3 \times 3 = 9 3^{2} = 9$

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Example 2
Is 16 a perfect square?

Try it out! Create a square using 16 Math U See Manipulative Blocks.
Record your model in Part 1 of your Lesson 8 Interactive Notes.

When you are ready, click the solution button below to compare your solution.

Solution

**The content below should appear when the solution button is clicked.

A perfect square can be created using 16 units.

As you can see, there are 4 units aligned horizontally across the top of the array, with another 4 units aligned top to bottom vertically along the array. This means that 16 is a perfect square of 4.

$4 \times 4 = 16 4^{2} = 16$

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Calculating Perfect Squares
To calculate a perfect square, multiply a number by itself.
For example to calculate the perfect square of 8, multiply 8 times 8.

$8 \times 8 = 64 8^{2} = 64$
The perfect square of 8 is 64.

The more you work with squares, the better you will get to know them.
Some of the most common perfect squares are listed in the table below.

Complete the table in Part 1 of your Lesson 8 Interactive Notes.

Exponent	Perfect Square
$1^{2}$
$2^{2}$
$3^{2}$
$4^{2}$
$5^{2}$
$6^{2}$
$7^{2}$
$8^{2}$
$9^{2}$
$10^{2}$
$11^{2}$
$12^{2}$

When you are ready, click the solution button below to compare your solution.

Solution

**The content below should appear when the solution button is clicked.

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Square Roots

A square root is one of two identical factors that when multiplied form a perfect square.

Think of it this way.
The root of a plant is where the plant begins under the ground. It is the base.
The root of a perfect square is the base that you start with. It is the number that is multiplied by itself to produce the perfect square.

Square roots are recorded using the radical sign (

\sqrt{}

Parts of a Radical
Use the diagram below to label and define the parts of a radical in your Lesson 8 Interactive Notes.

A radical is the symbol used to represent roots.

The index is always 2 for a square root, however, there is no need to record a 2 beside the radical sign. It is implied that the root is a square root when no index is recorded. The number that is squared is multiplied by itself one time, for a total of two equal groups of the root.

To calculate a square root you need to determine the factor that when multiplied by itself produces the radicand.

Example
$\sqrt{36}$ To calculate the square root of 36 you need to determine what number multiplied by itself will produce a product of 36.
$6 \times 6 = 36 \sqrt{36} = 6$
The square root of 36 is 6.
6 multiplied by itself produces a perfect square of 36.

Taking time to learn the common square roots above will make it easier to calculate many square roots.

Can a square root be negative?

The radicand can not be negative, however, the root of a positive radicand can be negative. When a number is squared it is multiplied by itself, thus for a negative root, the radicand becomes positive.

negative x negative =positive

It is impossible to square an integer and produce a negative product.

For this reason, a positive radicand can have two possible square roots.
The Principal Root of a number is its positive root.
The Secondary Root of a number is its negative root.

**It is only necessary to provide both the positive and the negative root if you are asked to provide all possible roots for a number. For most problems, you only need to provide the positive root.

Example
Consider $\sqrt{25}$

$5 \times 5 = 25 - 5 \times - 5 = 25$

5 and -5 are both possible square roots of 25.

Try it out!
Calculate all possible square roots for each value below.

1) $\sqrt{81}$
2) $\sqrt{49}$
3) $\sqrt{100}$
4) $\sqrt{225}$
5) $\sqrt{400}$

When you are ready, click the solution button below to compare your solution.

Solution

$1) \sqrt{81} 9 \times 9 = 81 - 9 \times - 9 = 81 9 o r - 9 2) \sqrt{49} 7 \times 7 = 49 - 7 \times - 7 = 49 7 o r - 7 3) \sqrt{100} 10 \times 10 = 100 - 10 \times - 10 = 100 10 o r - 10 4) \sqrt{225} 15 \times 15 = 225 - 15 \times - 15 = 225 15 o r - 15 5) \sqrt{400} 20 \times 20 = 400 - 20 \times - 20 = 400 20 o r - 20$

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Negative in Front of a Radical

What does the negative sign (-) in front of a radical mean, such as $- \sqrt{64}$ ?

When a negative sign is placed in front of a radical, you are asked to calculate the opposite of the root of the radicand.

Example
$- \sqrt{64} 8 \times 8 = 64$
The square root of 64 is 8.
The opposite of 8 is -8.
$- \sqrt{64} = (- 8)$

Try it out!
Evaluate each expression.

$1) - \sqrt{4} 2) - \sqrt{9} 3) - \sqrt{121}$

When you are ready, click the solution button below to compare your solution.

Solution

Compare your solutions to those shown below.
$1) - \sqrt{4} 2 \times 2 = 4$

The opposite of 4 is -4.
$- \sqrt{4} = (- 2)$
$2) - \sqrt{9} 3 \times 3 = 9$ The opposite of 9 is -9.
$- \sqrt{9} = (- 3)$
$3) - \sqrt{121} 11 \times 11 = 121$ The opposite of 11 is -11.
$- \sqrt{121} = (- 11)$

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**Instructor Notes – Text will appear when button is clicked

How to Support Students:

The following activities can be used to support students that need additional practice or enrichment on square roots and perfect squares.

Which one doesn’t belong?
- Present students with a set of numbers.
- Three of the numbers should be perfect squares, one is not
- Students must identify the number that is not a perfect square.
Matching
- Create a matching game. Students will match a perfect square to its root.
BINGO
- Create BINGO cards with a mix of roots and perfect squares.
- Students mark matches on their cards as they are called.
- The first student with BINGO wins.

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