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# Manhattan Algebra 1 Tutoring

### Module 1: Numbers

#### Topic A: Squares and Square Roots

Lesson Objective: Perform the four arithmetic operations using radical terms that are the same and different and write the result in its simplest form.

Squares

To square a number, just multiply it by itself.

Example: What is 3 squared?

“Squared” is often written as a little 2 like this:

This says “4 Squared equals 16”

(the little 2 means the number appears twice in multiplying, so 4×4=16)

Square Root

A square root goes the other direction:

3 squared is 9, so a square root of 9 is 3

A square root of x is a number r whose square is x:

r is a square root of x

Here are some more squares and square roots:

Negatives

We discovered earlier that we could square negative numbers:

Example: (−3) squared

(−3) × (−3) = 9

And, of course 3 × 3 = 9 also.

*So the square root of 9 could be −3 or +3*

Example: What are the square roots of 25?

**(−5) × (−5) = 25**

**5 × 5 = 25**

So the square roots of 25 are −5 and +5

**The Square Root Symbol**

This is the special symbol that means “square root” it is sort of like a tick and actually started hundreds of years ago as a dot with a flick upwards. It is called the radical and always makes mathematics look important!

and we say, “square root of 9 equals 3”

**Example:** What is √25?

25 = 5 × 5, in other words when we multiply 5 by itself (5 × 5) we get 25

So the answer is: √25 = 5

But wait a minute! Can’t the square root also be −5? Because (−5) × (−5) = 25 too.

Well the square root of 25 could be −5 or +5.

But when we use the radical symbol √ we only give the positive (or zero) result.

**Example:** What is √36?

**Answer:** 6 × 6 = 36, so √36 = 6

**Perfect Squares**

The Perfect Squares (also called “Square Numbers”) are the squares of the integers.

**Calculating Square Roots**

It is easy to work out the square root of a perfect square, but it is really hard to work out other square roots.

**Example:** what is √10?

Well, 3 × 3 = 9 and 4 × 4 = 16, so we can guess the answer is between 3 and 4.

Let’s try 3.5: 3.5 × 3.5 = 12.25

Let’s try 3.2: 3.2 × 3.2 = 10.24

Let’s try 3.1: 3.1 × 3.1 = 9.61

…

Getting closer to 10, but it will take a long time to get a good answer!

At this point, I get out my calculator and it says:

3.1622776601683793319988935444327

But the digits just go on and on, without any pattern.

So even the calculator’s answer is only an approximation

**Note:** numbers like that are called Irrational Numbers if you want to know more.

**A Fun Way to Calculate a Square Root**

There is a fun method for calculating a square root that gets more and more accurate each time around:

a) start with a guess (let’s guess 4 is the square root of 10) around

b) divide by the guess (10/4 = 2.5)

c) add that to the guess (4 + 2.5 = 6.5)

d) then divide that result by 2, in other words, halve it. (6.5/2 = 3.25)

e) now, set that as the new guess, and start at b) again

And so, after 3 times around the answer is 3.1623, which is pretty good, because:

3.1623 x 3.1623 = 10.00014

Now … why don’t you try calculating the square root of 2 this way?

**How to Guess**

What if we have to guess the square root for a difficult number such as “82,163” … ? In that case we could think “82,163” has 5 digits, so the square root might have 3 digits (100×100=10,000), and the square root of 8 (the first digit) is about 3 (3×3=9), so 300 is a good start.

**Surds**

When we can’t simplify a number to remove a square root (or cube root etc.,) then it is a surd.

**Example:** √2 (square root of 2) can’t be simplified further so it is a surd

**Example:** √4 (square root of 4) can be simplified (to 2), so it is not a surd!

Have a look at some more examples:

The surds have a decimal that goes on forever without repeating, and are Irrational Numbers.

**Simplifying Square Roots**

To simplify a square root: make the number inside the square root as small as possible (but still a whole number):

**Example:** √12 is simpler as 2√3

Get your calculator and check if you want: they are both the same value!

**Here is the rule:** when a and b are not negative

√(ab) = √a × √b

**Example:** simplify √12

12 is 4 times 3: √12 = √(4 × 3)

**Use the rule:** √(4 × 3) = √4 × √3

And the square root of 4 is 2: √4 × √3 = 2√3

So √12 is simpler as 2√3

**Example:** simplify √8

√8 = √(4×2) = √4 × √2 = 2√2

(Because the square root of 4 is 2)

**Example:** simplify √18

√18 = √(9 × 2) = √9 × √2 = 3√2

*It often helps to factor the numbers (into prime numbers is best):

**Example:** simplify √6 × √15

First we can combine the two numbers:

√6 × √15 = √(6 × 15)

Then we factor them:

√(6 × 15) = √(2 × 3 × 3 × 5)

Then we see two 3s, and decide to “pull them out”:

√(2 × 3 × 3 × 5) = √(3 × 3) × √(2 × 5) = 3√10

**Fractions**

There is a similar rule for fractions:

**Example:** simplify √30 / √10

First we can combine the two numbers:

√30 / √10 = √(30 / 10)

Then simplify:

√(30 / 10) = √3

**Example:** simplify

√20 × √5

√2

See if you can follow the steps:

√20 × √5

√2

√(2 × 2 × 5) × √5

√2

√2 × √2 × √5 × √5

√2

√2 × √5 × √5

√2 × 5

5√2

**Example:** simplify 2√12 + 9√3

First simplify 2√12:

2√12 = 2 × 2√3 = 4√3

Now both terms have √3, we can add them:

4√3 + 9√3 = (4+9)√3 = 13√3

**Principal Square Root**

So if there are really two square roots, why do people say √25 = 5 ? Because √ means the principal square root … the one that isn’t negative! There are two square roots, but the symbol √ means just the principal square root.

**Example:**

The square roots of 36 are 6 and −6

But √36 = 6 (not −6)

The Principal Square Root is sometimes called the Positive Square Root (but it can be zero).

**Plus-Minus Sign**

± is a special symbol that means “plus or minus”,

so instead of writing:

w = √a and w = −√a

we can write:

w = ±√a

In a Nutshell, When we have: r2 = x, then: r = ±√x

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#### Craig Selinger

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