or the fractions can be converted to floating point form and then added like this:
Furthermore, if an expression is an algebraic fraction and the numerator and the denominator are products of factors then the above sorting convention applies to the numerator and to the denominator. For example the expression on the left should be simplified to the one on the right:
The only exception is when the 1 is used as a placeholder as in this example (otherwise the numerator would be blank):
so no simplification is possible.
Click here for more information on invisible brackets.
a (b + c) = a · b + a · cThe quantity a can be a + or a − sign in which case you are distributing a 1 or a −1. Be aware that:
before simplifying after simplifying a + b + (c + d) a + b + c + d a + (−b) a − b (a − b) + (−c + d) a − b − c + d a − (−b + c) a + b − c − (−b + c) b − c
Click here for details on adding like terms and more examples.
Evaluating an exponential is often considered to be a function called the power function.
Here is an example of each of these cases:
Click for more information on functions in general and the eleven built-in functions of the Algebra Coach.
Here is an example of a simplification where the exponential notation for the square root is useful:
The exponential function can be written two ways: using exponential notation and using function notation. Exponential notation is the standard:
Reciprocals can be written in two ways: using fraction notation and using exponential notation. Fractional notation is the standard:
Among other things, exponential notation for reciprocals is useful for seeing the connection between factoring and finding common denominators.
In words, the non-fraction becomes a factor in the new numerator. This is just a special case of multiplying two fractions. Here’s why:
Similarly, a fraction divided by a non-fraction should be replaced by a single fraction, like this:
In words, the non-fraction becomes a factor in the new denominator. This is just a special case of dividing two fractions. Here’s why:
In words, the two fractions a/c and b/d multiply to give a new fraction whose numerator is ab and whose denominator is cd. Click for more information on multiplying common fractions and multiplying algebraic fractions. Here is an example:
This simplification is really an example of the associative law of multiplication, which states that it doesn’t matter in what order we multiply and divide the factors. This simplification has all kinds of consequences:
In words, the division of the fraction c/d by the fraction b/a is replaced by the multiplication of the fraction c/d by the reciprocal a/b. Then this multiplication of two fractions is carried out as described above. Click for more information on dividing common fractions and dividing algebraic fractions. Here is an example:
Here is another example:
This is a very useful result: The result of dividing two fractions with equal denominators is a fraction consisting of just the two numerators.
b 0 = 1
b 1 = b
1 x = 1
In other words any quantity raised to the power −1 is just the reciprocal of that quantity. The fraction form, 1/b, on the right side of this property is usually considered to be the simpler form but the exponential form, b−1, on the left is useful for seeing, for example, the connection between factoring and finding common denominators.
Here are some examples:
Click here for more information and examples.
This property is used to replace the complicated exponential on the left side by the simpler exponential on the right side. Here are some examples:
Note: The last example shows that raising a quantity to the 1/3 power is the same as taking the cube root of that quantity.
Note that this simplification applies only to products or quotients. It does NOT apply to sums or differences! Click here for more information and examples.
In words, b m/n may be thought of as the nth root of the mth power of b or as the mth power of the nth root of b.
Click here for more information.
You can use this property to simplify and replace the expression on the left by the one on the right. The right side is considered to be simpler than the left side because an exponentiation on the left side is replaced by a multiplication on the right side. Here are two examples:
(The second example first used the logarithm of a product property.)
You can use this property to simplify and replace the expression on the left by the one on the right. The right side is considered to be simpler than the left side because a reciprocal on the left side is replaced by a negative on the right side. Here is an example:
This can be simplified further if we recognize that 128 and 4 are both 2 raised to different powers:
10 x = 33can be solved for x by taking the (base 10) logarithm of both sides, like this:
log (10 x ) = log (33)According to the fourth row in the above table the left side of the equation then simplifies like this:
x = log (33)and the right side can then be evaluated.
To derive the simplifications, refer to the triangles. For example the first triangle has been set up so that θ = arctan(x). Pythagoras’ theorem then gives the length of the third side. The first two simplifications result from getting the values of sin(θ) and cos(θ) for that triangle. The other simplifications are derived the same way.
|These function compositions often occur when we solve an equation by doing the same thing to both
sides of the equation. For example suppose that in the triangle shown to the right we want to find the tangent of the angle
θ without actually finding θ.
This can be done by noticing that
θ = arcsin (0.4)and taking the tangent of both sides of this equation. This gives:
tan (θ) = tan (arcsin (0.4))
and it can now be evaluated.
arccos(sin(θ)),They all simplify to π/2 radians − θ (or to 90° − θ in degree mode). To prove this notice that these 4 statements apply to the triangle shown:
Now simply apply the operations indicated by the arrows:
sin(−α) = −sin(α)
cos(−α) = cos(α)
tan(−α) = −tan(α)
sin(x + 270°) = cos(x + 180°) = −sin(x + 90°) = −cos(x).These identities are an important part of the algorithms used by the Algebra Coach and by calculators and computers to evaluate the sin, cos, and tan functions.
sin2(α) + cos2(α) = 1
1 − sin2(α) = cos2(α)
1 − cos2(α) = sin2(α)
The square root of any negative number can be simplified by expressing it as a multiple of the imaginary unit i. Here are two examples:
In engineering fields where the symbol i already represents the electric current, the symbol j is often used to denote the imaginary unit.
Click here for more information.
e i π = −1.Click here for more information.