# Download Cohomology Operations: Lectures by N.E. Steenrod. by N. E. Steenrod, David B. A. Epstein PDF By N. E. Steenrod, David B. A. Epstein

Written and revised by means of D. B. A. Epstein.

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Example text

When an exponent is a positive integer, it indicates the number of times a number (the base) is multiplied by itself. For example, 2 to the power of 3 is the same as 23 = 2 · 2 · 2 = 8. , the exponent, 2⁄3, isn’t When an exponent is a rational number, like in the expression equivalent to the number of times 8 is multiplied by itself. To evaluate this expression, you should write it as or , which is equivalent to and , respectively, with the numerator acting as a positive integer exponent and the denominator acting as the index of the radical.

From there, still working on the right-hand side of the coordinate plane, move over 1 and up 1, over 1 and up 1. Continue this pattern to complete the right side of the graph; use symmetry to complete the left half of the graph (see Figure 3-3). 5 4 3 2 h ( x )= |x| y 1 Figure 3-3: The parent absolute value function. 0 –1 –2 –5 –4 –3 –2 –1 0 x 1 2 3 4 5 Cubic functions Cubic functions are defined by third-degree polynomials, so the highest exponent on any one variable is 3. The parent cubic function is p(x) = x3, which is an odd function.

Multiply through by the common denominator, 4, to eliminate the fractions altogether. Then solve like normal by combining like terms and isolating x. Here’s the math: g Solve . The answer is x = –3/4, –1/2. So you have two absolute value terms? Just relax and remember that absolute value means the distance from 0, so you have to consider all the possibilities to solve this problem. In other words, you have to consider and try four different possibilities: both absolute values are positive, both are negative, the first is positive and the second is negative, and the first is negative and the second is positive.