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Exponents: Squares and Square Roots
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Skills Application: Exponents: Squares and Square Roots
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The math term “squaring” has been with us since Medieval times. In those days, paper and ink were expensive and students often did their math work in the dirt on the ground. They did multiplication with stones. To multiply 4 by 4, stone pebbles were arranged in a square formation. Students made 4 rows of 4 pebbles, then counted all the pebbles to get the result, 16. By doing math this way the ideas of numbers arranged into shapes was developed. The term square root simply referred to the side length of these pebbles squares used for multiplication.

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In mathematics, we use exponents as a way to abbreviate repeated multiplication.

Example:

The exponent form of 6 x 6 is 62. There are two parts to exponent notation:

(1) the base which is the number 6

(2) the exponent which is the small elevated number 2

The base tells you what number is being multiplied and the exponent tells you how many times this number is used as a factor. In the above example the base is 6 and the exponent is 2.

Square

When the exponent is 2 the result is called the square. Recall that 62 means 6 x 6. That is, 6 appears as a factor two times. The base is the variable 6. You also have an x2 button on your calculator to speed up squaring. Locate it on your calculator and use it to solve the questions in the practice problems.

Square roots

We say that 2 is a square root of 4 because 2 x 2 = 4. What is interesting about the numbers 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, . . . ? Do you recognize them? These numbers are the squares or second powers of the counting numbers,1, 2, 3, 4, 5, 6, 7, 8, 9, 10, . . . :

12 = 1
22 = 4
32 = 9
42 = 16
52 = 25, and so on. 1, 4, 9, 16, 25, . . . are called perfect squares.

Ancient Greek mathematicians called certain numbers square numbers or perfect squares because they could be represented by a square array of dots.

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The number of dots along the side of the square was called the root or origin of the square number. We call it the square root. When you square this number, or multiply it by itself, you obtain the original number.

Example

(a) The square root of 16 is 4, because 42 or 4×4 = 16

(b) The square root of 64 is equal to 8 because 82 or 8×8 = 64

Square Root

The sign √ is used to indicate the square root

$\sqrt{16}$
= 4          read it as “square root of 16 is 4”

$\sqrt{169}$
= 13      read it as “square root of 169 is 13”

Finding the Square root

How do you find the square root of any number? Using a calculator is the easiest way. Find out where the button is on your calculator and whether it is a second function 2ndF or not. Find it and use it to figure out the practice problems.

Get a \$20-or-so dual-display scientific calculator, even if you cant take it with you into the exam hall. A calculator is also a learning tool, Technology Use is one of the 9 Essential Skills you cant do without. Trying out different functions on a calculator will further your understanding and help with building a bigger mental picture. It builds links between numbers and algebra. It is important to try alternative calculations or approaches, to sketch & label angles, tracking changes to units of measure, visualizing to cross-link the concepts of area and numbers. ### A Short Review

Squaring a number or a unit of measure means multiplying it by itself.

52 = 5 · 5

m2 = m · m

Whole numbers, fractions, decimals can all be squared.

Square rooting a number means dividing to find the number that was squared in the first place.

$\sqrt{25}$
= 5, because 5 was squared to get 25

$\sqrt{100}$
= 10, because 10 was squared to get 100

Whole numbers, fractions, decimals can all be square rooted. ### Worksheet: Level 1

The operations used are clearly specified. Only one type of mathematical operation is used in a task.

### Worksheet: Level 1 Sample Questions

Example:

What multiplication is meant by:

1.  212 = ___ · ___

2.  4.52 = ___ · ___

### Worksheet: Level 1 Answer Key

1.  212 = 21 · 21

2.  4.52 = 4.5 · 4.5 ### Worksheet: Level 2

Tasks involve one or two types of mathematical operation. Few steps of calculation are required.

### Worksheet: Level 2 Sample Questions

Example:

Calculate the following:

1.   6.82 = _____
2.
$\sqrt{2.8}$
= _____

Further Examples:

Calculate the following:

1. 0.0232 = _____
2. $\sqrt{2.04}$
= _____

### Worksheet: Level 2 Answer Key

1. 6.82 = 6.8 x 6.8 = 46.24
2. $\sqrt{2.8}$
= 1.673 ### Worksheet: Level 3

Tasks require a combination of operations. Several steps of calculation are required.

### Worksheet: Level 3 Sample Questions

Example:

Calculate the following:

5 m2 + 3 m2 =
5 m2 – 3 m2 =
5 m2 · 3 =
5 m2 ÷ 3 =

Further Examples:

Calculate the following:

1.  7 m2 + 4 m2 =

2.  10 in2 – 2 in2 =

3.  3 mm2 · 4 =

4.  8 yd2 ÷ 2  =