Math: Finding Square Root or Cube Root from Prime Factorization

The math tutor continues to appreciate prime factorization for all it yields.

Let’s imagine you need to determine the square root of a number without a calculator. This challenge is part of the curriculum for local high school students.

Example:  Determine if each number can be square rooted (to a whole number).  If so, find its square root.

a)  540
b)  576

Before tackling the above problem, let’s dissect a number we know to be a perfect square.

Example:  Confirm, by prime factorization, that the square root of 900 is 30.

Solution:  We recall that a prime number is one that cannot be divided into smaller numbers, then break 900 down into primes:

900=10×90=(2×5)(9×10)=(2×5)(3x3x2x5)=2x5x3x3x2x5

Rearranging, we get

900=2x2x3x3x5x5=(2x3x5)(2x3x5)

We notice 900 can be broken into two identical groupings like so:

900 = (2x3x5)(2x3x5)

Therefore, the square root of 900 is 2x3x5=30

We now know what to seek:  if a number is square rootable, its prime factorization can be organized into two equal groups.  The square root is simply the product of one of the groups.

Back to our example:

Determine the whole number square root (if it exists) of the following:

a) 540
b) 576

Solution:

a)  First we break 540 into primes:

540=10×54=(2×5)(6×9)=(2×5)(2x3x3x3)=2x2x3x3x3x5

With only one 5 in the prime factorization, we can’t separate it into two equal groups. 540 doesn’t have a whole number square root.

b)  2 and 4 both go into 576.  Without a calculator, you either do it mentally or else use long division.  To get started, just break it in half:

576=2×288=2(2×144)=2(2x12x12)=2(2x(3×4)(3×4))=2x2x3x4x3x4

Since we have only multiplication here, we can add and rearrange brackets at will. However, with mixed operations we wouldn’t be able to do so:)

Rearranging, we get

576=2x2x3x3x4x4=(2x3x4)(2x3x4)

Clearly, the prime factorization of 576 is separable into two equal groupings of 2x3x4. 2x3x4 = 24, so the square root of 576 is 24.

If its prime factorization can be separated into three equal groupings, the number is a perfect cube:

Example:  Confirm that 9261 is a perfect cube.

Solution:  We’ll break this one down using short division.  Since 9+2+6+1=18, we know 9 divides into it:


Since 1+0+2+9=12, we know 3 divides into it (because 3 divides into 12).


Since 3+4+3=10 (which 3 doesn’t divide into), and 343 doesn’t end in 5 or 0, the next number to try is 7:

So we see that we can break down 9261 as follows:

9261=9(1029)=9(3×343)=9(3x7x49)=(3×3)(3x7x7x7)=3x3x3x7x7x7

Rearranging, we separate the prime factorization into three equal groupings:

9261=3x3x3x7x7x7=(3×7)(3×7)(3×7)

Therefore, 9261 is a perfect cube with cube root=3×7=21.  The cube root of 9261 is 21.

Once again, short division was used to break down 9261.  For more explanation about that very handy technique, please check future posts:)

Jack of Oracle Tutoring by Jack and Diane, Campbell River, BC.

Math: Introduction to Polynomials

As a math tutor, you deal with polynomials half the weeks of the year.

A polynomial is an expression in which the variables can have only positive, whole-number exponents.  Examples of polynomials are 3x7-12x5+1 or  -2x – 12.

In a polynomial, terms are separated by plus or minus signs.  Therefore, 7x – 12 has two terms, whereas  -13x7yz2 has only one.

Polynomials are often named by how many terms they have, as follows:

7x2                   One term: monomial

3x – 12             Two terms: binomial

-4x4 + 3x -5     Three terms: trinomial

A polynomial with more than three terms is just called a polynomial.

Consider the following monomial:

-11x5

-11 is called the coefficient.  x is called the variable.  5 is the exponent.

The degree of a polynomial is the highest exponent found in one of its terms.  For example, 5x3 has degree 3. The trinomial

x7 + 3x3 – 12

has degree 7.

The constant term of a polynomial is the term with no variable attached. In 3x – 12, the constant term is -12.

There are two facts about polynomials that might be a little surprising:

Fact 1:

A constant term has degree zero.  Reason: x0= 1 by definition. The result:

3x0 = 3(1) = 3.

We just write 3, but the degree of the term is still zero.

Fact 2:

If the coefficient is not written, it is 1.  Reason: x5 = 1x5.

There are a couple of finer points, but the above is good for a start.  Much more will be said about polynomials in future posts.

Jack of Oracle Tutoring by Jack and Diane, Campbell River, BC.

Biology: Alternation of Generations

I’ve always found this topic a bit tricky.  As a biology tutor, it’s time to give my own explanation.

Alternation of generations refers to the idea that the life cycle of a moss, for instance, comprises two distinct phases:  one in which the cells are haploid (n), and the other in which they are diploid (2n).  Haploid means they have half the number of chromosomes; diploid means they have the full possible number.

Although all plants have it, moss provides a good example of alternation of generations.  People may not notice as they walk over it, but moss looks different depending on the time of its cycle.  Most often you are probably seeing the gametophyte (n) generation – likely the green, soft carpet you imagine in forests or bogs.  However, if the carpet has brown stalks rising from it, you might well be walking over the sporophyte (2n).

Let’s enter the cycle at the spore stage.  A spore is a haploid (n) cell, often borne on the wind, which can land and grow into a new individual – specifically, a new gametophyte.  If the spore is male, it grows into a sperm-producing gametophyte; if female, an egg-producing one.  In wet conditions, sperm swim from the male stalks to the female ones.  (This is why moss needs dampness at least some of the time.)  Like you’d expect, the union of sperm with egg produces a zygote.  While the sperm and egg were each n, the zygote is 2n, since it receives n chromosomes from the sperm and n from the egg.

The zygote starts to divide, eventually becoming a sporophyte found on top of what was the female gametophyte (which is where the sperm joined the egg).  Cells in the mature sporophyte undergo meiosis, a special kind of division in which each cell produced receives only one chromosome from each set of two available.  Whether it receives the chromosome the sperm brought, or the one from the egg, is determined by chance:  hence the variability of sexual reproduction.  Either way, these new cells are haploid (n):  in fact, they will mature into spores.  The spores are released when it’s dry and windy, and the life cycle begins anew.

I’ll be mentioning more about this topic in future posts;  this was just a toe in the water:)

Jack of Oracle Tutoring by Jack and Diane, Campbell River, BC.

Math: when to use Permute, when to use Choose

With counting problems, the math tutor needs to explain the uses of permute and choose.

Permute and choose are counting functions.  Found on any scientific calculator, they look similar to nPr and nCr, respectively.

You use permutation (nPr) to count the number of arrangements.

Example:  Eight people are posing in a row for a photograph.  How many ways can they be positioned?

Solution:  Assuming they are all in the photograph, the answer is 8P8, which works out to 40 320 ways.

Example, Continued:  What if only 5 of the 8 people are going to be in the photograph?

Solution:  Now, the answer is 8P5:  6720 ways.

You use combination (nCr) to count the number of assortments – that is, the number of ways you can select a set of r objects from a set of n objects.

Example:  How many ways can you select three crayons from a box of 12?

Solution:  12C3, or 220 ways.

With choose, the order of the selection doesn’t matter – just which items are selected. With permute, the order of the items matters.

Both permute and choose are  without replacement – which means that once an object is placed or selected, it can’t “come up” again.  This only makes sense:  a person can’t show up in two different places in the same photograph.  Similarly, if you are using only one deck of cards, you can’t receive the queen of hearts, then receive it a second time in the same hand.

Other counting strategies will be discussed in future posts:)

Jack of Oracle Tutoring by Jack and Diane, Campbell River, BC.

Math: How to read logs

Logarithms – logs, for short – are a perennial for the math tutor.

What is a log?  You see the button on your calculator, but only in high school (if ever) do you likely find out what it means.

First, we need some review:

Now, we show a simple log equation:

This log equation is read as follows:

The exponent you would put on 2 to make 8 is 3.

Of course, it’s true:

When you see a log with no base written, the base is 10.

There is a great deal more to say about logs – which we will, in future posts.

Enjoy the sun:)

Jack of Oracle Tutoring by Jack and Diane, Campbell River, BC.

Math: Inequalities and Number Lines

As a math tutor, you’ll likely need to sort out a few basics about inequalities.

Generally, people prefer equations to inequalities.  Perhaps it’s for the reason that people like one definite answer to a problem:  for instance, “x=5”.  An inequality gives a set of valid answers, such as “x can be anything less than -1”.  I guess we’ll just have to get used to it:)

More problematic – and more fixable – is that many people don’t know which sign is which. We can sort that out right now:

<   means less than

>   means greater than

Some examples:

10 < 12

20 > 8

Consider the number line:

Both less than and greater than actually refer to placement on the number line.  “Less than” means “to the left of”, while “greater than” means “to the right of”.  Of course, 10 is greater than 5, which is written 10 > 5.  Notice that on the number line, 10 is to the right of 5.

Similarly, 4 is less than 7.  In math, you write 4 < 7.  Notice that on the number line, 4 is to the left of 7.

It follows, of course, that -10 is less than -5.  After all, on the number line, -10 is to the left of -5.  Therefore,

-10 < -5

People often have trouble with the idea that -10 < -5.  They point out that 10 seems bigger than 5, so why would -10 be less than -5?

The answer is that “less than” doesn’t refer to number size; it means “to the left of”.  On a number line, -10 is to the left of -5.  Therefore, -10 < -5.  Similarly, 5 being to the left of 10, we write 5 < 10.

The practical issues of solving and graphing inequalites will be explored in future posts:)

Jack of Oracle Tutoring by Jack and Diane, Campbell River, BC.

Biology: Matters of the Heart

When you tutor Biology 12, you need to discuss the heart.

As is commonly known, the heart is a muscular pump.  Its beat pushes the blood through the vessels to every part of the body.

The human heart has two sides.  The pulmonary side receives oxygen-depleted blood from the vena cava (the big, final veins), then pumps it through the pulmonary artery to the lungs to get re-oxygenated.  The systemic side receives oxygenated blood from the lungs (via the pulmonary vein), then pumps it through the aorta which divides to service every part of the body.

Each side of the heart has two chambers:  a reception chamber and an output chamber. The reception chamber is called the atrium or the auricle.  The output chamber is called the ventricle.

One practical problem every plumbing system needs to prevent is backflow.  The heart prevents backflow using valves.  Each side of the heart has two valves:  one between the atrium and the ventricle, then one between the ventricle and the outgoing artery.

The valve between a ventricle and its outgoing artery is called a semilunar valve.  The left semilunar valve prevents blood flowing backward from the aorta into the left ventricle.  The right semilunar valve prevents backflow from the pulmonary artery into the right ventricle.

The valve between an atrium and a ventricle has several possible names.  It can be known, generally, as an atrioventricular valve.  However, each atrioventricular valve has its own unique name(s) as well.  For instance, the right atrioventricular valve is also known as the tricuspid.  The left atrioventricular valve has two specific names:  it can be called the bicuspid or the mitral valve.

The heart’s pace is ultimately decided by the medulla oblongata, located in the brain stem. However, when the medulla oblongata chooses not to interfere, the heartbeat is self-governed from the SA node, which is set to around 70 bpm.  If the SA node is damaged, the AV node can step in, but it creates a heartbeat of only 40-60 bpm. Hence, a pacemaker might be needed.

Hope you enjoyed this heartfelt discussion,

Jack of Oracle Tutoring by Jack and Diane, Campbell River, BC.

Source:  Inquiry into Life, Eleventh Edition, Sylvia S. Mader.  McGraw-Hill:  2006.

Math: The Statement and the Contrapositive

For a few years, this topic fell from view.  As a math tutor, I’m glad to see it back.

To a person studying logic, the statement “p implies q” also means “if p, then q”.  It can also be written

p→q

Example of a statement:

If a minute has passed, then sixty seconds have passed.

p and q, by themselves, might be called assertions.  Therefore, in the above statement, “a minute has passed” is an assertion.  So is “sixty seconds have passed.”

To form the contrapositive of a statement, you reverse its order, then negate both parts of the statement:

If sixty seconds have not passed, then a minute has not passed.

In logic notation, you negate an assertion by writing a line above it:


It follows that the construction of the contrapositive is

I’m told that, in general terms, the contrapositive is the logical equivalent to the statement itself. From what I’ve seen myself, I’ve no cause to doubt that assertion:)

There are other logical derivatives of a statement: the converse and the inverse, to name a couple. I’ll discuss them in future posts:)

Jack of Oracle Tutoring by Jack and Diane, Campbell River, BC.

Math: the meaning of a negative exponent

As a math tutor, you’ll need to remind students about this exponent law.

Most exponent laws people find pretty straightforward.  I’ll likely cover them in a future post. However, this particular one deserves its own; most people just don’t like it.  Let’s discover it’s really not so bad.

Rule:

Example:

Notice the fraction version:

Example:

A consequence of this rule is that a negative power, if on the bottom, can be moved to the top and made positive:

Example:

The rule needs to be followed literally. Like most rules in math, it often doesn’t lead to the final answer. Rather, it normally occurs as a step on the way to the final answer. Apply it exactly, then proceed!

Jack of Oracle Tutoring by Jack and Diane, Campbell River, BC.

Math: Evens and Odds

What’s the difference between evens and odds?  When you’re a math tutor, you might need more than the obvious answer.

Everyone knows that 0, 2, 4, 6….are even, whereas 1, 3, 5, 7, 9….are odd.  Negative numbers can also be even or odd:  -8 is even, whereas -7 is odd.  Formally, the mathematical definition of “even” is as follows:

2p, p is any integer.   The integers are  {….-3,-2,-1,0,1,2,3….}.

The definition of odds:

2q+1, q is any integer.

Therefore, 2(-11) = -22 is even.  On the other hand, 2(-8) + 1 = -15 is odd.

An even can’t divide (without a remainder) into an odd:  every even number has 2 as a factor, and 2 won’t divide into an odd number (by definition).

On the other hand, an odd can divide into an even.  3, for instance, divides into 12.

Here’s a fun fact:  the square of an odd is odd.

Proof:  assume the odd is 2t + 1.  Then its square is (2t + 1)2.  Multiplying by the foil method (see my post on foil here):

(2t + 1)2=(2t + 1)(2t + 1)=4t2 + 4t + 1.

Notice:

4t2+ 4t + 1 = 2(2t2 + 2t) + 1.

By definition, 2(2t2+ 2t) + 1 is an odd number:  it is written in the form 2(integer) + 1.

The nuances of even and odd can reveal some surprising discoveries, as we’ll see in future posts:)

Jack of Oracle Tutoring by Jack and Diane, Campbell River, BC.

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