The Easy Way to Simplify Square Roots

In a first year algebra class, students will encounter problems involving simplifying square roots. But many times the teacher simply doesn't explain the process by which this is accomplished in a manner that students can understand. I will explain a way that will make simplifying square roots easier for students of any ability level.

The first method I use when teaching students how to calculate square root is to look to see if the number is a perfect square first. I suggest students memorize the perfect squares from 1 to 25 as follows: 1, 4, 9, 16, 25, 49, 64, 81, 100, 121, 144, 169, 196, 225, 256, 289, 324, 361, 400, 441, 484, 529, 576, 625. So if you are asked to calculate the square root of 441, you automatically know the answer is 21 or -21, since a negative number times another negative number yields a positive number.

For non-perfect squares or larger numbers that you are unsure of, if the number is a perfect square, I suggest using a factor tree. For example, suppose you want to simplify a square root of 48. It is not a perfect square since 6 times 6 equals 36 and 7 times 7 equals 49. No whole number times itself equals 48. So break it down into factors. I always suggest trying to find a perfect square as one of the factors and in this case, 16 times 3 equals 48 and 16 is a perfect square. Remember: Since you are dealing with square root, the factors are also square root. So the square root of 48 equals the square root of 16 times the square root of 3. Three is a prime number so you cannot break down 3 any farther using a factor tree. We know that the square root of 16 is 4, so the answer is 4 times the square root of 3.

Another more difficult example---say we need to find the square root of 2025. With such a large number, most people won't know if this is a perfect square, so use the factor tree. Know that any number ending in 5 is divisible by 5. So 5 times 405 is 2025. But 405 can be broken down into factors, using the same rule, therefore 5 times 81 is 405. Now we have the square root of 5 times the square root of 5 times the square root of 81. Notice then that the square root of 5 times the square root of 5 equals the square root of 25. Now this problem becomes simple because you notice we have two perfect squares here, 25 and 81. The square root of 25 is 5 and square root of 81 is 9. So the answer is 5 times 9, which is 45 and -45, since -45 times -45 equals 2025. This problem is actually a perfect square, but if you do not recognize it as such, you can use the factor tree method I just described.

When dealing with the square root of a negative number, imaginary numbers come into play. The square root of -1 equals an imaginary number denoted as "i". So in the above problems if we had the square root of -48, you have square root of -1 times square root of 3 times square root of 16. The answer is 4i times the square root of 3, and -4i times square root of 3. If we have the square root of -2025, the answer is simply 45i and -45i.

I love math but for those that don't, this is pretty funny!

It's easy to remember the formula for volume of a cylinder if you know the area of a circle..  A cylinder is basically a tube, which has two bases, which are circles and a height.

So the area of a circle is Pi(r^2)

Think of the cylinder as a bunch of CD's stacked up top of each other until you reach the total height of the cylinder.  So it's obvious to multiply the area of the circle by the height to get the volume of the cylinder.

Remember the change of base formula when solving a logarithm that is not base 10.

Log(base 3) 11 = Log 11/Log 3

It's easy to see how this comes about when rewriting the log in exponential form and then solving for y.

y = log(base 3) 11

3^y = 11

Take the log of both sides to get

log(3^y) = log 11

ylog 3 = log 11

y = log11/log3

Instead of having to rewrite as an exponential equation and solving, simply use the change of base formula. In general

Log(base b)a =  Log a/Log b

When calculating limits, try to factor the numerator and denominator first. Then simplify before evaluating the limit. If there is a radical in the denominator, rationalize it first.

Make sure when doing sided limits, when taking the limit from the left (-) and from the right (+), they must be the same or the limit does not exist.

Students taking a course in basic statistics will learn about many types of probability distributions. The most widely used and most important is the normal probability distribution. What are the characteristics of this distribution and how is it graphed?

The normal distribution is a continuous distribution with numerous applications. We need to learn about some of the properties of this distribution.

The normal distribution is defined in terms of its mean and standard deviation. The graph will give one some idea of the main features of this normal distribution. The graph of the normal distribution is called the normal curve. It's also called the bell-shaped curve since it very much resembles a bell.

The curve is symmetrical about the vertical line that extends up from the mean. The highest point on this graph is about the mean. The standard deviation controls the amount of spread in the curve. It is very close to the x-axis at mean + 3 times standard deviation and mean - 3 times standard deviation. This implies that when the standard deviation is small, the curve is less spread out and more spread out when the standard deviation is large.

Summarizing the important properties:

1. The curve is bell-shaped with the highest point at the mean.
2. The curve never touches the x-axis.
3. The curve is symmetrical about a vertical line through the mean.
4. The points between the curve cupping upward and downward occur at the mean plus or minus the standard deviation.

The empirical rule is a rule used for the normal distribution, and all other symmetrical, bell-shaped distributions. It states the approximately 68% of the data values fall within one standard deviation of the mean. Approximately 95% of the data values fall within two standard deviations of the mean. Approximately 99.7% of the data values fall within three standard deviations of the mean.

This guide should help assist students having difficulty understand the basics of the normal probability distribution and its graph.

Some Properties of the Binomial Distribution

The main focus of a binomial experiment is to find the probability of r successes in n trials. How are these probabilities calculated? How is mean and standard deviation of a binomial distribution calculated? How is a binomial distribution displayed graphically?

Suppose you wish to flip a fair coin three times. What is the probability of obtaining two heads in the three tosses? This is an example of a binomial experiment, with probability of success 0.5. We could list the possibilities as {HHH, HHT, HTH, THH, HTT, TTH, THT, TTT}. You notice from the sample space that the probability is 3 out of 8, or 3/8. But most times it's much too time consuming to list all possibilities. So there is a formula that we can use:

P(r) = n!/[r!(n - r)!]* p^rq^(n-r)

where n = number of trials
p = probability of success on a trial
q = probability of failure on a trial
r = random variable representing the number of successes out of n trials

Using the coin tossing example above, P(0) = 1/8, P(1) = 3/8, P(2) = 3/8, P(3) = 1/8.

To graph a binomial distribution, place the r values on the horizontal axis. Next, place the P(r) values on the vertical axis. Then construct a bar over each r value. Extend these bars to r - 0.5 to r + 0.5. The height of each bar must be P(r).

In the above example the bar widths would be -0.5 to 0.5, 0.5 to 1.5, 1.5 to 2.5, and 2.5 to 3.5. You cannot have a 0.5 of a success but to have a bar, there needs to be a width, so you add and subtract 0.5 from each value of r.

To help describe the graph of the binomial distribution, the mean and standard deviation are very helpful. For this distribution, the mean is np and the standard deviation is the square root of npq.

In our example with the coins, the mean is 3(0.5) = 1.5, and the standard deviation is square root of 3(0.5)(0.5) = 0.87.

This simple guide should help students understand some properties of the binomial distribution including calculating probabilities, graphing, and computer the mean and standard deviation.

Computing Quartiles and Box-and-Whisker Plots

In statistics, many times data is summarized by its mean and standard deviation. But in some instances, such as when the data is heavily skewed or bimodal, we may be more interested in relative position of data instead of precise values. In such cases percentiles are used. Special percentiles frequently used, particular in box-and-whisker plots, are known as quartiles.

Recall that the median of a set of data is the middle value when ordered in ascending or descending order. Since half the data fall above the median and half the data fall below the median, the median is the 50th percentile. In general, the Pth percentile of a distribution is the value where P% of the data falls at or below it and the rest of the data falls above it.

We can expand on the topic of percentile by introducing special percentiles known as quartiles, which are percentiles that divide the data into fourths. The first quartile, known as Q1 is the 25th percentile, the second quartile, known as Q2 is the median, and the third quartile, known as Q3 is the 75th percentile. It is important to know how to find Q1, Q2, and Q3 in order to draw a box-and-whisker plot.

To compute quartiles, first order the data from smallest to largest. Next, find the median. The first quartile is the median of the lower half of the data. The third quartile is the median of the upper half of the data. The interquartile range, Q3 - Q1, gives the spread of the middle half of the data.

Example: Find Q1, Q2, Q3, and the interquartile range of the following set of data.

5, 10, 8, 9, 10, 2, 5, 6, 11, 15, 8

First, we order the data from smallest to largest: 2, 5, 5, 6, 8, 8, 9, 10, 10, 11, 15. The median is 8 since half 2, 5, 5, 6, 8 falls at or below it and 9, 10, 10, 11, 15 falls at or above it. The median of the lower half of the data is 5, which is Q1. The median of the upper half of the data is 10, which is Q3. The interquartile range is Q3 - Q1, which is 5.

The quartiles are used with the maximum and minimum values of a data set to create a box-and-whisker plot. These are very useful to describe a data set. To make a box-and-whisker plot, draw a vertical scale that includes the highest and lowest data values. Then mark Q1, the median and Q3. Draw a box around Q1 and Q3. Draw a line through the box where the median is. The whiskers are then drawn, which are vertical lines from Q3 to the maximum value and from Q1 to the lowest value.

In the data set above, the box would be around 5 and 10. The median line is through 8 in the box. The whiskers are drawn from Q1 to the lowest data value of 2, and from Q3 to the largest data value of 15.

This guide should help students learn the basics about percentiles, quartiles and constructing box-and-whisker plots.

Scatter Diagrams and Linear Correlation

A scatter diagram is a graph where the data points (x, y) are plotted on a rectangular coordinate system, where x is the horizontal axis and y is the vertical axis. Scatter diagrams are used in studies of correlation and regression in two variables.

After a scatter diagram is made for a set of data, a line is drawn through the points. This line is known as the "line of best fit". But how do we determine which is the "best" line through a set of points? It is the line that comes closest to each point in the scatter diagram. The "least squares line", which can be computed by hand using the data values, or more easily by a computer, will be the best fit line. This line will contain the mean of the x value and the mean of the y value. In fact, the coordinate is (x mean, y mean).

Sometimes the data is dispersed in a way that there is no "best" line. If the points are a poor fit to any line, it makes no sense to try to find a line of best fit. When the points are scattered in a way that there is not a "good" fit, then there is no linear correlation between the x and y values. Picture many randomly scattered points, almost as if looking into the sky at a bunch of stars. There will be very little if any linear correlation between the points. If the points are scattered in a way that you can visually see where a line would go or the points almost form a line, then there will be linear correlation and strong linear correlation the more the points form a straight line.

The measurement that determines the strength of linear association between variables is known as the "sample correlation coefficient r". Also known as Pearson's correlation coefficient, named after statistician Karl Pearson.

The correlation coefficient is a measurement between -1 and 1. A correlation coefficient of -1 means there is perfect negative linear correlation between x and y. On the scatter diagram the points would form a perfect line with negative slope. A correlation coefficient of 1 means there is perfect positive linear correlation between x and y. On a scatter diagram the points would form a perfect line with positive slope. The closer  r is 1, the stronger the positive correlation, and the closer r is to -1, the stronger the negative correlation. Basically this means that the closer r is to 1 and -1, the more the line describes the relationship between the variables.

In linear correlation, the explanatory variable is x and the response variable is y. These are also known as the independent and dependent variables, respectively. The value for r can be calculated by hand from the data pairs using a tedious formula or can be easily calculated using a computer.

This guide should help assist students learning the basics about scatter diagrams and linear correlation.

when working with polar coordinates, remember that there are several ways to represent the same point.  For example, the point (2, Pi) is the same as the point (-2, 0), (-2, -Pi), (2, 0).

You can check this by converting the polar coordinates to rectangular coordinates.
If the polar coordinate is represented as (r, A) where r is the radius and A is the angle, then
x = rcos(A),   y = rsin(A)

Using the above example,

x= 2cos(Pi)
y= 2sin(Pi)  converts to (-2,0) in rectangular coordinates

Likewise using the same formula for the polar coordinates (-2, 0), (-2, -Pi), (2, 0) also convert to (-2, 0) in rectangular coordinates.