Simplifying radicals: variables with coefficients

Tutoring math, simplifying radicals constitutes one of the most difficult topics for high school students.  The math tutor offers a step-by-step approach which continues here.

In my previous post, I mentioned how simplifying the square root of a variable to a power is slightly different from simplifying the square root of a number. Let’s review quickly:

Example 1: Simplify √48

Solution:

Step 1: Factor 48 into the biggest perfect square that goes into it, times the number it goes in:

48=16×3 so √48=√16√3

Step 2: Take the square root of 16.

√48=4√3.

Example 2: Simplify √x21

Solution:

Step 1: Realize that √x21=√x20√x

Step 2: Realize that √x20=x10 (Since x10x10=x20)

Therefore, √x21=√x20√x=x10√x

Now, let’s perform the two processes side by side:

Example 3: Simplify √28x15y8

Step 1: First, separate the radical into a convenient product.

√28x15y8=√28√x15√y8

Step 2: Tackle each part separately.

28=4×7; x15=x14x;

√28√x15√y8=√4√7√x14√x√y8=(2√7)(x7√x)(y4)

Step 3: Recollect all the simplified terms to the front.

(2√7)(x7√x)(y4)=2x7y4√7x

Terms that have been “rooted out” go in front of the radical so that they are clearly not in it. The terms behind a radical sign are meant to be in it. Such is the convention used almost universally in the math world.

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

Math: simplifying radicals, Part II

Tutoring math, you need to explain this every semester.  The math tutor draws the distinction between handling the numbers and the variables.

Looking back to my earlier post on radicals, you’ll get the basic idea. Just for a quick review, suppose you have √50. You need to break it down to √25√2. Then, take the square root of 25 to get 5. Your simplified answer is 5√2.

Note that you need to break the number into a perfect square times whatever else. If a perfect square doesn’t factor into the number, you can’t simplify it. For example, √35 does not simplify. You can write it as √7√5, but neither 7 nor 5 is rootable (into a whole number). Therefore, we leave it √35.

What about √x10? Well, the answer is x5. The reason is that (x5)(x5)=x10. When you multiply the two terms, you add their exponents. Therefore, √x14=x7.

Now, let’s look at √x15. How do we simplify it? We break it down as √x14√x. Now we realize that √x14=x7. Our simplified answer is x7√x.

Next time, we’ll tackle a situation with both numbers and variables.

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

Math: measuring angles: degrees, minutes, and seconds

Tutoring math, you don’t see this very often these days.  The math tutor recalls learning it in Math 12.

Let’s imagine you have 32.65 degrees.  In the “old days”, it might have been stated as 32 degrees, 39 minutes.  It was gotten this way:

0.65×60=39 minutes.

The “minutes” come from the decimal part of the degree measurement.  Since there are 60 minutes in a degree, take the decimal part of the degrees measurement and multiply by 60.  It’s the same as realizing that 0.25 hours (which is, of course, a quarter of an hour) is 15 minutes:

0.25×60=15 minutes.

What if, after multiplying by 60, you still get a decimal?  That resulting decimal can be converted to seconds by mutliplying by 60 once again.

Example:  Convert 56.87 degrees to degrees, minutes, seconds.

First, use the decimal part to get the minutes:

0.87×60=52.2 minutes.

Now, use the decimal from the minutes to get seconds:

0.2×60=12 seconds.

56.87 degrees is 56 degrees, 52 minutes, 12 seconds.  You can write it  56°52’12”.

To my knowledge, mariners still use minutes and seconds.  High school math students may use them in the future.  When something is taken out of the math curriculum, it often returns years later….

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

Math: practical calculator hints

Tutoring math, you notice the various calculators in use.  The math tutor reveals a couple of important differences among them.

When I was in high school, I used a Texas Instruments scientific calculator.  What model it was I don’t know, but it was virtually the same as today’s TI-30XA.

Back then, the calculators I knew of were all reverse entry:  when you wanted to take the square root of a number, you’d enter the number first, then tap the square root key. You’d take sin, cos, or tan of an angle the same way:  by entering the number first, then the function.  My Texas Instruments TI-30XA and Casio fx-260SOLAR are both reverse-entry.

Among the calculators I see people using nowadays, reverse-entry ones constitute the minority.  More of my students use forward entry calculators:  to get the square root of a number, they tap the square root key, follow by entering the number, then press the “equals” button.  The same goes for sin, cos, and tan:  they enter the function, then the number, and finally “equals”.

Most people know which type of calculator they have and how to use its basic functions. However, some may not realize how their calculator handles brackets.  For example, 4(7+3) means 4x(7+3) to a human.  However, neither my TI-30XA, nor my CASIO fx-260SOLAR recognizes that the brackets mean multiplication.  Each requires you to enter 4x(7+3) in order to recognize you intend to multiply.

On the other hand, my forward-entry SHARP EL-520W does accept 4(7+3) as meaning 4x(7+3); it will give 40 when you enter 4(7+3)=.  However, it will not do so when you enter (7+3)4=.

There are many models of calculator in use today.  A student will do well to experiment with his or hers, to be sure of how it interprets brackets and/or expects functions to be entered.  While many are similar, you can’t always safely assume that your new one works just like your old one.

More differences among various models of calculator will be explored in future posts:)

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

Math: the “fraction button”

Tutoring math, you’ll become aware of the fraction button.  As a math tutor, I don’t recommend using it.  Nonetheless, people commonly do….

Looking at a scientific calculator, you’ll likely see a button that looks like this: abc. It’s the fraction button.

If you enter 5abc7 + 1abc3 =, you’ll probably get 1r 1r 21. This means 1121. Now press shift (or 2nd F or just 2nd, whichever your calculator has) abc; you’ll see 22r21, which means 2221.

The fraction button saves you from having to get a common denominator to add or subtract fractions. Therefore, using it can save a lot of effort – especially for someone who’s weak at times tables.

My advice: don’t use it. In high school math, you’ll encounter algebraic fractions, which the scientific calculator can’t handle. You’ll have to do them “the old way”, by hand. Therefore, this math tutor recommends keeping in touch with how to do so.

The fraction button didn’t exist when I was in school. I watched, fascinated, as a student showed it to me in the late 90s. Normally, I don’t use it; however, even some study guides recommend its use now. In certain contexts, I guess using it could make sense.

Hope this helps:)

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

English: two common sentence faults

Tutoring English, you deal with these issues constantly.  In this post, the English tutor sheds light on a couple of common sentence faults.

You burnt the cake she is furious.      fused sentence

You burnt the cake, she is furious.     comma splice

Neither the fused sentence nor the comma splice is correct.  Both are found commonly in everyday writing.  The comma splice is even found in published writing.

The problem with both the fused sentence and the comma splice is that each produces a run-on sentence. A run-on sentence is incorrect.  A simple way to identify one is that its subject changes or is restated without a semicolon or conjunction in between. There are other punctuation fixes besides the semicolon, but a comma by itself won’t work.

You burnt the cake she is furious.           Subject change from you to she.
I went to Bill’s party, I had a great time.    I is restated.

Facing a fused sentence or a comma splice, the same fixes can be used.  One solution is to add a conjunction before the second subject; another is to use a semicolon.

You burnt the cake, so she is furious. The conjunction so fixes the sentence.
You burnt the cake; she is furious. The semicolon correctly separates the two
complete thoughts.
I went to Bill’s party and I had a great time. The conjunction and, probably overused in writing, corrects the sentence in this case.

Fused sentences and comma splices are best gotten rid of. Changing habits always seems easier at the beginning of the school year:)

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

Source: Harbrace Handbook for Canadians, sixth edition. 2003: Nelson Education Ltd.

Math: prime and relatively prime

Tutoring math, these definitions don’t come up often enough. The math tutor offers this brief read on them.

A factor of a number divides into it with no remainder.  For example, 5 is a factor of 15; 7 is a factor of 56.

A prime number has exactly two factors:  1 and itself.  Notice that 2 is prime, having only 1 and 2 as factors.  7 is also prime.  9 is not prime, since it has three factors:  1, 3, and 9 all divide into it with no remainder.  10 is not prime either.  1 is not prime, since it only has one factor:  1.

Numbers that are relatively prime don’t have to be prime, but they share no common factor except for 1.  12 and 5 are relatively prime, for instance.  10 and 15 are not relatively prime, since they have 5 as a common factor.

An application of relatively prime numbers is fraction reduction.  If the numerator (the top number) and the denominator (the bottom number) are relatively prime, the fraction is reduced.  Reduced might also be referred to as in lowest terms.

By that rationale, 2418 is not in lowest terms because 24 and 18 are not relatively prime.   In particular, they share the common factor 6.  Dividing 6 out of both, we get 43. Since 4 and 3 are relatively prime, the fraction is now reduced.

Hope this helps:)

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

English: Active vs Passive

With another school year upon us, we resume tutoring math, sciences, English, etc.  Tonight:  a post about English.

Active voice and passive voice are both easy to understand.  Behold:

Active:  Bob presented the speech.

Passive:  The speech was presented by Bob.

Since the passive voice gives a gentler rendering of an event, most people prefer it. Consider the following:

Active:  I broke your ruler.

Passive:  Your ruler got broken.

Which telling would you more likely expect?

While the passive voice is preferred by everyday people, the active voice is preferred by people who mark writing assignments.  Specifically, the active voice gives a livelier, more honest telling of an event.  Such is the opinion of most English academics nowadays.

Hope this helps.  By the way:  welcome back to school!  All the best this academic year from Oracle Tutoring by Jack and Diane, Campbell River, BC.

Source: Harbrace Handbook for Canadians, sixth edition. 2003: Nelson Education Ltd.

Math: hexadecimal numbers

Tutoring math, this is an interesting concept.  Its common application is in computer science.

You might have seen numbers like c6 or e9.  Depending on the context, these may be hexadecimal (aka hex) numbers.  The hexadecimal system is base 16.  It uses the following notation:

decimal hex
0 0
1 1
2 2
3 3
4 4
5 5
6 6
7 7
8 8
9 9
10 a
11 b
12 c
13 d
14 e
15 f

Since base 16 requires numbers that would be two digits in base ten, the number system is simply expanded to 15 using letters.

Continuing the logic from my previous post, we see that c6 means

12×161+6×160=12×16 + 6×1=192 + 6=198.

So, c6 in hex is 198 in decimal.

Since 162=256, you can represent any number from zero to 255 with two digits in hex. ff=255, as follows:

ff=15×161+15×160=240+15=255.

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