Topic 8.2 (Rational Expressions II)
x-n = 1 or 1 = xn If a factor has a negative exponent in the numerator,
xn x-n move it to the denominator and change the sign of the
exponent.
If a factor has a negative exponent in the denominator, move it to the
numerator and change the sign of the exponent.
Never leave an answer with a negative exponent in it.
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Sample problem: 7-2 · 73 change to 1 · 73 = 73 = 7 Ans.
72 72
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Sample Problem: 50 Change the denominator from negative exponents to
10-2 + 5-2 positive exponents.
50 This is a complex fraction with 50 in the numerator and
1 + 1 the other two fractions in the denominator.
102 52
50 Add the denominators and invert and multiply.
1 + 1
100 25
50
1 + 4
100 100
50 = 50 x 5 = 50 · 100 = 1000 Ans.
5 1 100 1 5
100
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Changing a number to scientific notation is changing the number to this form:
One digit to the left of the decimal times 10 raised to an exponent.
(the exponent represents the decimal point)
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Sample problem: 73901 changes to 7.3901 X 104
The decimal point has been moved four places from the right to the left and that means a positive exponent of 10.
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Sample Problem: .00004003 changes to 4.003 X 10-5
The decimal point has been moved five places from the left to the right and that means a negative exponent of 10.
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A problem like this: 8 - x Factor out a negative 1 in the numerator.
x - 8
-(x - 8) Reduce the fraction and the answer is -1.
x - 8
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When dividing fractions, invert the fraction on the right of the division sign and use the steps of multiplication.
When multiplying fractions, factor the numerator and denominator of each fraction completely, and then cross out, on a one to one basis, a common factor in the numerator and denominator of the same fraction or the numerator and denominator of a different fraction.
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Sample: (x + 3)(x - 2) ÷ (x + 3)(x - 4) Factor numerator and denominator
x2 - 4 x2 - 16 completely.
(x + 3)(x - 2) x (x + 4)(x - 4) Invert the fraction on the right and multiply.
(x + 2)(x - 2) (x + 3)(x - 4) Cross out the common factors.
x + 4 Ans.
x + 2
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Complex fractions (a fraction in the numerator and the denominator) must be changed to one term in the numerator and the denominator, before performing the division.
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Sample: 3/x + 5/y This is a complex fraction because there is a fraction in the
4/x - 7/y numerator and a fraction in the denominator. There are
two fraction in the numerator and the denominator.
These must be combined and changed so there is only
fraction (a monomial in the numerator and the
denominator.)
3y + 5x This is the addition and subtraction Algorithm.
xy a/b ± c/d = ad ± bc
bd
4y - 7x
xy Now invert and multiply.
3y + 5x x xy Cross out the common factors.
xy 4y - 7x
3y + 5x Ans.
4y - 7x
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To add and subtract fractions, a lowest common denominator must be used. Each fraction must be changed, so the lowest common denominator is shown in each denominator.
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Sample: y - -9 The problem has the same denominator, so combine the
y2 - 81 y2 - 81 the numerators, factor completely, cross out common
factor (reduce) and you have your answer.
y - -(9) = y + 9
y2 - 81 (y + 9)(y - 9)
1/(y - 9) Ans.
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Sample: 8 - 4 Factor each denominator first.
x2 + 14x + 49 x2 - 49
8 - 4 The LCD is all the different factors of the
(x + 7)2 (x + 7)(x - 7) denominator, raised to the largest power
shown. Raise all fractions to the LCD
LCD = (x + 7)2(x - 7)
8(x - 7) - 4(x + 7) The first fraction needs to be multiplied (numerator
(x + 7)2(x - 7) and denominator) by (x - 7).
The second fraction needs to be multiplied
(numerator and denominator) by (x + 7)
8x - 56 - 4x - 28 Multiply and simplify.
(x + 7)2(x - 7)
4x - 84 If the numerator can be factored, and one of the
(x + 7)x - 7) factors reduced with a factor in the denominator,
do it, if not that is the answer.