June 2011

The Expansion of (a + b)ⁿ
If   $n$   is any positive integer, then

$(a + b)^n = a^n + {_nC_1}a^{n - 1}b + {_nC_2}a^{n - 2}b^2 + \, \cdots \, + {_nC_m}a^{n - m}b^m + \, \cdots \, + b^n$
 

Where
${_nC_m}$ = combination of n objects taken m at a time.
 

Special Products
1.   $(x + y)(x - y) = x^2 - y^2$

2.   $(x + y)^2 = x^2 + 2xy + y^2$

3.   $(x - y)^2 = x^2 - 2xy + y^2$

Properties of Logarithm

1. If   $y = a^x$,   then   $\log_a y = x$.   ← Definition of logarithm
2. $\log_a xy = \log_a x + \log_a y$
3. $\log_a \dfrac{x}{y} = \log_a x - \log_a y$
4. $\log_a x^n = n \log_a x$
5. $\log_a a = 1$
6. $\log_a 1 = 0$
7. $\log_{10} x = \log x$   ←   Common logarithm
8. $\log_e x = \ln x$   ←   Naperian or natural logarithm
9. $\log_y x = \dfrac{\log x}{\log y} = \dfrac{\ln x}{\ln y}$   ←   Change base rule
10. If   $\log_a x = \log_a y$,   then   $x = y$.
11. If   $\log_a x = y$,   then   $x = {\rm antilog}_a \, y$.

Laws of Exponents (Index Law)
1. $x^n = x \cdot x \cdot x ... \, (n \text{ factors})$

2. $x^m \cdot x^n = x^{m + n}$

3. $(x^m)^n = x^{mn}$

4. $(xyz)^n = x^n \, y^n \, z^n$

5. $\dfrac{x^m}{x^n} = x^{m - n}$

6. $\left( \dfrac{x}{y} \right)^n = \dfrac{x^n}{y^n}$

7. $x^{-n} = \dfrac{1}{x^n}$   and   $\dfrac{1}{x^{-n}} = x^n$

8. $x^0 = 1$,   provided   $x \ne 0$.

9. $(x^m)^{1/n} = (x^{1/n})^m = x^{m/n}$

10. $x^{m/n} = \sqrt[n]{x^m}$

11. If   $x^m = x^n$,   then   $m = n$   provided   $x \ne 0$.
 

Problem 338
The two 12-ft beams shown in Fig. 3-16 are to be moved horizontally with respect to each other and load P shifted to a new position on CD so that all three reactions are equal. How far apart will R₂ and R₃ then be? How far will P be from D?
 

Problem 337
The upper beam in Fig. P-337 is supported at D and a roller at C which separates the upper and lower beams. Determine the values of the reactions at A, B, C, and D. Neglect the weight of the beams.
 

Problem 336
The cantilever beam shown in Fig. P-336 is built into a wall 2 ft thick so that it rests against points A and B. The beam is 12 ft long and weighs 100 lb per ft.
 

Problem 335
The roof truss in Fig. P-335 is supported by a roller at A and a hinge at B. Find the values of the reactions.
 

Problem 334
Determine the reactions for the beam loaded as shown in Fig. P-334.