Suppose you wish to multiply a number, variable or expression a repeated number of times. Exponents allow you to write these products in a simpler form. For example, 5 ∙ 5 ∙ 5 ∙ 5 = 54, y ∙ y ∙ y ∙ y = y⁴ , (x - 3) ∙ (x - 3) ∙ (x - 3) ∙ (x - 3) ∙ (x - 3) ∙ (x - 3) = (x - 3)⁶ . Each of the simplified expressions are called exponential expressions. The number being raised to the exponent is known as the base and the exponent is known as the power. But how do we deal with the concept of negative exponents?

The general rules for positive exponents hold true for negative exponents. If x is an integer and n ≠ 0, then x^-n = 1/xⁿ. To see why this holds true, multiply the equation by xⁿ to get x^-n ∙ xⁿ = 1. Remember when multiplying exponential expressions, you add the exponents, so -n + n = 0 and x⁰ = 1.

You can think of this as the reciprocal of x^-n and changing the exponent from negative to positive. Generally speaking, you want to make sure you have all positive exponents in the expression in simplest form.

Examples: Write each expression with only positive exponents.

3^-4 , (2a)^-2 , 5x^-3

3^-4 = 1/(3⁴) = 1/81
(2a)^-2 = 1/(2a)² = 1/4a²
5x^-3 = 5/(x³)

Note when raising a fraction to a negative exponent, take the reciprocal of the fraction and change the exponents to positive.

Example: Simplify the expression using only positive exponents.

(x/y)^-3

Take the reciprocal first then change the negative -3 to 3.
(y/x)
(y/x)³

Then apply the rule for raising an exponent to an exponent (remember y is y¹ and x is x¹). Therefore, (x/y)^-3 = y³/x³.

Example: Simplify the expression using only positive exponents.

4x^-3/(x¹⁰)

There are two ways to solve this. You can apply the rule for dividing exponential expressions and subtract the exponents.

4x^-3/(x¹⁰) = 4x^-(3-10) = 4x^-13

Then change the negative exponent to positive by changing x^-13 to 1/x¹³. The simplified form is 4/x¹³. You can change the negative exponent to positive first to get 4/(x³x¹⁰), then apply the rule for multiplying exponential expressions to get 4/x¹³.

Example: Simplify the expression using only positive exponents.

(x⁵x⁶)/(x^-4)

Keep the common base x in the numerator and add the exponents to get x¹¹/x^-4
Subtract the exponents to get x^{(11 + 4)} = x¹⁵. You could also use the rules for negative exponents to get x⁵x⁶x⁴ = x¹⁵ .

The examples above should help students who are having trouble understanding the concept of negative exponents.

When dealing with polar coordinates, it's easy to convert to rectangular coordinates


x = rcos(angle)

y=rsin(angle)

If the polar coordinate (r,angle) is (4, pi/2)

The rectangular coordinates are

x = 4cos(pi/2) = 0

y = 4sin(pi/2) = 4

If you are given rectangular coordinates, you can convert to polar coordinates


x^2 + y^2 = r^2

tan(y/x) = angle

So if you have the rectangular coordinates (2, 3)

The polar coordinates are

2^2 + 3^2 = r^2

13 = r^2

sqrt(13) = r

tan(3/2) = angle

angle = 56.3 degrees

In geometry two triangles are similar if the corresponding  sides are in proportion. If  two angles are the same, the are also similar. If two angles are the similar, in fact, all three angles must be the same.

For example

In triangle ABC, AB = 10, BC = 15, AC = 18

In triangle DEF, DE = 20,  EF = 30, DF = 36

These triangles are similar because corresponding sides are in proportion

10/20 = 15/30 = 18/36.  All three fractions simplify to 1/2

Try another example.

Triangle ABC,  AB = 5, BC = 12, AC = 15

Triangle DEF,  DE = 15, BC = 30, AC = 45

For the triangles to be similar,  5/15 = 12/30 = 15/45

Simplify each fraction to get 1/3 = 2/5 = 1/3.. Notice this is a false statement, so the triangles are not similar


Another example dealing with the angles

Triangle ABC, <A = 45,  <B = 89

Triangle DEF,  <E = 89,  <F = 46

<C = 180 - (45 + 89) = 46

<D = 180 - (46 + 89) = 45

<A corresponds with <D,  <B corresponds with <E and <C corresponds with <F.  Since the angles are congruent, the triangles are similar.

 Although we can solve quadratic equations by completing the square, sometimes this process is cumbersome and a bit difficult. There is a simpler way to solve quadratic equations through the use of the quadratic formula.The quadratic formula is found by completing the square of a quadratic equation in the form ax^2 + bx + c = 0, a > 0.


ax^2 + bx + c = 0
ax^2 + bx = -c (Subtract c from both sides of the equation)
x^2 + (b/a)x = -c/a (Divide both sides of the equation by a)
x^2 + (b/a)x + b^2/(4a^2) = -c/a + b2/(4a^2) (Complete the square by taking half of (b/a) , squaring it and adding to both sides)
[x + (b/2a)]^2 = -c/a + b^2/(4a^2) (Factor the perfect square trinomial on the left side)
[x + (b/2a)]^2 = (b^2 – 4ac)/4a^2 (Get common denominator of 4a^2 on the right side)
x + (b/2a) = +/- √[(b^2 – 4ac)/4a^2] (Take the square root of both sides (use the square root property))
x + (b/2a) = +/- √(b^2 – 4ac)/2a (Simplify the square root on the right side √(4a^2) = 2a)
x = [-b +/- √(b^2 – 4ac)]/2a (Subtract b/2a from both sides)


The result is known as the quadratic formula

Remember that division by zero does not exist.  Just as addition undoes subtraction, multiplication undoes division. 

For example..  5 + 2 = 7. To undo the addition of 2 to 5, subtract 2 from 7.    7 - 2  = 5

8 - 5 = 3.  To undo the subtraction, add 5 to 3 to get  3 + 5 = 8

Now try multiplication to undo division.   6 divided by 3 = 2.  To undo, multiply 2 and 3 to get 6.

This works, but in the next example, it will not.

4 times 0 = 0. If you follow what was done in the examples above, 0 divided by 0 should equal 4, but that is obviously not true.

Through basic mathematics, we are taught that we could not evaluate the square root of a negative number because the square root of a negative number is not a real number. For example, the solution to x² = -2 is not a real number solution because the square of a real number is always positive. In fact, when trying to find the square root of a negative number on a calculator, you will get a "syntax error", or simply "error" message. Therefore, to solve such equations, the number "i" was created such that i² = -1. The imaginary number "i" is defined as √-1 = "i". The imaginary numbers are part of another number system known as the complex number system.

To solve the square root of a negative number or expression, we use the same rules for multiplying and dividing radicals. We write the square root of a negative as "i" times a real number. In general, for any real number a > 0, √-a = i√a.

Examples: Simplify each of the following radicals.

1. √-16 = √-1√16
= i∙4 (Replace √-1 with i)
= 4i

2. √-5 = √-1√5
= i√5 (Replace √-1 with i)
=√5i

Note that i√5 and √5i are the same. Generally, we write the imaginary number with the radical after the "i". The reason is because it's easy to confuse √(5i) and √5i.

3. √-24 = √-1√24
= √-1√4√6 (Break √24 down into √4√6)
= i∙2√6 (Simplify √4 and Replace √-1 with i)
=2i√6

4. √(-36/81) = √-36 /√81 (Rewrite as a division of two radicals)
= (√-1√36)/√81
= i√36/√81 (Replace √-1 with i)
= 2i/3 (Simplify 6/9 to 2/3)

In the previous examples, we showed how to simplify the square root of negative real numbers. We can extend this to simplifying the square root of monomials containing a negative real number and variables.

Examples:

1. √-49x²y⁴ = √-1 ∙√49∙√x²∙√y⁴ (Rewrite as product of radicals)
= i∙7xy² (Simplify all radicals and replace √-1 with i)
= 7i(xy²)

2. √-7x³y² = √-1∙√7∙√x³∙√y² (Rewrite as product of radicals)
= i∙√7∙x∙√x∙y (Simplify all radicals and replace √-1 with i)
= xyi√(7x)

We learned how to simplify the square root of negative numbers with the use of the imaginary number "i". Next we use the imaginary number to define complex numbers. A complex number is any number of the form a + bi where a and b are real numbers and √-1 = "i". Some examples are complex numbers are as follows:

4 - 13i, -12 + 27i, 3 - 3i√2 and (5/2) + (3/2)i.

We can add, subtract, multiply and divide complex numbers. To add and subtract complex numbers, combine the real parts and the imaginary parts.

Examples: Add or subtract the following complex numbers.
1. (3 + 4i) + (10 + 6i)
3 + 10 = 13 (Add the real parts)
(4 + 6)i = 10i (Add the imaginary parts)
34

Therefore, (3 + 4i) + (10 + 6i) = 13 + 10i.

2. (-14 + 3i) - (4 + 7i)
-14 - 4 = -18 (Subtract the real parts)
(3 - 7)i = -4i (Subtract the imaginary parts)

Therefore, (-14 + 3i) - (4 + 7i) = -18 - 4.

Recall the product rule when multiplying two positive real numbers. If a and b are positive real numbers, then √a√b = √(ab). When trying to apply the same rule to multiplying the square root of two negative numbers, we get a wrong result. For example, using the product rule for √-3√-27 gives us √(-3)(-27) = √81 = 9. But this is incorrect. What we need to do is write each radical in terms of "i". When doing this we get i√3 ∙ i√27 = (i2)√81 = (-1)(9) = -9.

Examples: Multiply the following complex numbers.

1. √-40√-10 = √-1√40√-1√10
= i∙√40∙i∙√10 (Replace √-1 with i)
= i2∙ √400 (Multiply the i's and use product rule for radicals to multiply √40√10)
= (-1)(20) (Replace i² with -1 and √400 with 20)
= -20

2. √-36√-9 = √-1√36√-1√9
= i∙6∙i∙3 (Simplify the radicals and replace √-1 with i)
= i2∙18 (Multiply the i's and multiply 6 and 3)
= (-1)(18) (Replace i2 with -1)
= -18

The distributive property of multiplication and FOIL that apply to real numbers also apply to complex numbers. Sometimes a problem is in the form (a - bi)(a + bi). These complex numbers are known as complex conjugates and are used in division where the denominator is a complex number. It's used because multiplying complex conjugates results in a real number and that removes the complex number from the denominator. It's the same idea as removing radicals from the denominator. Therefore, to divide complex numbers, multiply the numerator and the denominator by the complex conjugate.

This guide should help anyone having difficulty understanding imaginary numbers and how to perform mathematical operations on complex numbers.

What is a radical function? A radical function is a function that contains roots. Examples of radical functions are f(x) = √x, g(y) = 3√(2y + 3) and h(z) = √(3z - 5). One of the most common radical functions is the square root function. We will start with the most basic square root function, f(x) = √x. We are interested its graph and its domain and range. First, we'll find the domain of the function. Recall that only values for x ≥ 0 satisfy √x. If x < 0, we will not get a real number solution for √x.

To graph f(x) = √x, we'll set up a table of values.
For x = 1, f(1) = √1 = 1
x = 4, f(4) = √4 = 2
x = 9, f(9) = √9 = 3
x = 16, f(16) = √16 = 4

We would plot this on a rectangular coordinate system with the horizontal axis being x, the vertical axis being f(x). Note that the vertical axis on a rectangular coordinate system is generally the y axis. Notice we used values for x that are perfect squares. They are much easier to graph.

Note that we only used the principal square root for the values of f(x). The reason for this will be noted
after the graph. Recall that the square root has both positive and negative values (ie, √4 = 2 and -2). But the reason we did not use the negative values on the graph is because we are assuring that it is a function. If we graph the negative values, the graph is a parabola which opens to the right. This would not pass the vertical line test and would not be a function. Therefore the graph of the function f(x) = √x are only the values in the first quadrant.

Knowing the graph of the basic square root function, f(x) = √x, will help us graph many other radical functions. If h > 0, then f(x) = √(x +h) and f(x) = √(x - h) are the graphs of f(x) = √x moved to the left and right h units, respectively. For example, the graph of f(x) = √(x + 1) has an x-intercept is (-1, 0). When x = 0, f(x) = 1. The domain is x ≥ -1 because x < -1 will yield a negative under the radical and the square root of a negative is not a real number. You can plot several other points that makes √(x + 1) a perfect square. This graph is the graph of f(x) = √x moved 1 unit to the left. It is said to be translated 1 unit to the left. The graph of f(x) = √(x - 1), similarly, would be the graph of f(x) = √x moved 1 unit to the right.

We learned how to graph radical functions that are translations left and right of the square root function, f(x) = √x. Now we will learn how to graph radical functions that are that translations up and down of the square root function f(x) = √x. If k >0 , then f(x) = √x - k and f(x) = √x + k are the graphs of f(x) moved down k units and up k units, respectively. Therefore the graph of f(x) = √x + 3 is the same as the graph of f(x) = √x except it is translated 3 units up. Similarly, the graph of f(x) = √x - 3 is the same as the graph of f(x) = √x except it is translated 3 units down.

A function can have multiple translations. For example, the graph of f(x) = √(x - 2) + 1 is the same as the graph of f(x) = √x except it is translated 2 units right and 1 unit up. You will notice when graphing that there is vertical line through x = 2. The domain of the function is x ≥ 2. This shows that there are no points on the graph to the left of 2 on the x axis. The graph is in the first quadrant where both x and f(x) are positive.

The cube root function is a radical function defined as f(x) = ³√x. To graph the function we want to choose values for x that are a perfect cube. For example 1 is a perfect cube because 1(1)(1) = 1 and 8 is a perfect cube because (2)(2)(2) = 8.

To graph f(x) = ³√x, we'll set up a table of values.
Fox x = 1, f(1) = ³√(1) = 1
x = -1, f(-1) = ³√(-1) = -1
x = 8, f(8) = ³√(8) = 2
x = -8, f(-8) = ³√(-8) = -2
x = 27, f(27) = ³√(27) = 3
x = -27, f(-27) = ³√(-27) = -3

The function contains values in the first and third quadrants and no values in the second and fourth
quadrants. As with the square root function, there are translations with the cube root function. The graph of f(x) = ³√x - 3 is the same as the graph of this function is the same the graph of f(x) = ³√x except it is translated down 3 units.

This guide should help clear any confusion on the topic of radical functions and graphing radical functions.