(a+b)², because (a
+ b)² = a² + 2ab + b²!Addition and subtraction of powers
If you want to evaluate 6·2³ + 4·2³, you can do it in two ways:
1) 6·2³ + 4·2³ = 6·8 + 4·8 = 48 + 32 = 80, 2) 6·2³ + 4·2³ = 10·2³ = 80.
2³ is a definite number a; like 6a + 4a = 10a, 6·2³ + 4·2³ = 10·2³. However, such a combination by addition and subtraction is only possible, if you have equal powers such as 2³, that is, if the powers have an equal base and exponent. This Rule is frequently violated!
6·2³ + 4·3² can only be
evaluated in the form 6·8 + 4·9 = 84;
3² + 4² only as 9 + 16 = 25.
Only 3² + 4² = 5², no other sums of squares
equal squares! a²+b² (a+b)², because (a
+ b)² = a² + 2ab + b²!
Examples:
| 1) | 6a² + 4a³ - 2a² - a³ = 4a² + 3a³ | |
| 2) | 16x³ + 4x² + 3x - 12x³ + 6x² - x = 4x³ + 10x² + 2x | |
| 3) | 6a²b + 4ab² = 6a²b + 4ab² (a²b and ab² are not the same!) | |
| 4) | 10a²b + 8ab² - 6a²b - 4ab² = 4a²b + 4ab² | |
| 5) | 10a²bc 5 + 4a²b²c - 6a²bc = 4a²bc + a²b²c |
During multiplication and division, there may arise simplifications if the powers are not equal, but only either have the same base or the same exponent.
Multiplication and division of powers with equal base.
| a³·a² = (a·a·a)·(a·a) | I. (am)n = am+n |
a³ means 3 factors, a² means 2 factors, together you have then 5 factors, which is written a5.
Examples:
| 1. 34·38 = 312 | 2. (-2)³ ·(-2)4 = (-2)7 | 3. (1/2)²·(1/2)³ = (1/2)5 |
This rule is readily extended to more than 2 powers:
| 4. x3·x4·x5 = x12 | 5. am·an·am·ar = a2m+ n+r |
In order to evaluate the quotient a5/a3, the same reasoning applies, that is, you represent the powers as products of equal factors and simplify in the usual way:
| a5/a3 = (a·a·a·a·a)/(a·a·a) = a·a = a² | II. am/an = am-n |
The numerator has 5 factors, the denominator 3 factors, 3 factors cancel and you obtain a2.
We have assumed that m > n. If m < n, you obtain, for example,
| a3/a5 = (a·a·a)/(a·a·a·a·a) = 1/a·a = a-2 | in general: am/an = 1/an-m |
Examples:
| 1. x8/x5 = x3 | 2. (-3)10/(-3)8 = (-3)2 = 1/9 |
Since the rules for powers have the form of equations and left and right hand sides can be interchanged, you find easily their inverses (cf. Calculations with sums and differences )
Example: Put brackets around common factors:
a8 + a4 + a2 = a2·a6 + a2·a2 + a2 = a2(a6 + a2 + 1).
Multiplication and division of powers with equal exponents
| a3·b3 = a·a·a·b·b·b = ab·ab·ab = (ab)3 | III. an·bn = (ab)n |
Examples:
| 1. 22·32 = 62 | 2. 53·23= 103 = 1000 |
Expansion to several factors:
| 3. a4·b4·c4·d4= (abcd)4 | 4. 22·52·32= (2·5·3)2 = 302 = 900 |
Inversion: (ab)n = an·bn.
Examples:
| 1. (2a)³ = 2³·a³ = 8a³ | 2. (-2b)³ = (-2)³·b³ = -8b³ | |
| 3. (3ab)4 = 34·a4·b4 = 81a4·b4 | 4. (2a)³ + 2a³ = 8a³ + 2a³ = 10a³ |
a4/b4=(a·a·a·a)/(b·b·b·b)=(a/b)·(a/b)·(a/b)·(a/b)=(a/b)4. IV. an/bn = (a/b)n
Examples:
| 1. 82/42 = (8/4)2 = 22 = 4 | 2. (ab)2/b2 = (ab/b)2 = a2 |
Example:
| Inversion: | (a/b)n = an/bn | (2/3)³ = 2³/3³ = 8/27 |
Powers of powers
| (a3)4 = a³·a³·a³·a³ = a3·4 = a12. | V. (am)n = am·n |
Examples:
| 1. | ((1/2)²)³ = (1/2)6 = 1/64 | 2. | [(-a)3]5 = (-a)15 = -a15 |
Since (am)n = am·n and (am)n = am·n, then also (am)n = (an)m.
| Inversion: | am·n = (am)n = (am)n | Example: 23n = (23)n = 8n |
| I. | am·an=am+n | II. | am/an=am-n (m>n) | am/an=1/am-n (m<n) | |||||
| III. | (ab)n=an·bn | IV. | (a/b)n=an/bn | V. | (am)n=(an)m=am·n |
Given any problem involving powers, it is important to be clear about every step in a calculation. Besides the definition, you may only use the five rules above, since otherwise you may make mistakes. After some practice, you can naturally employ several of the rules simultaneously.
Examples:
| 1. | (23)4·(32)6·(52)4/(22)5·310·56 = (V) = 212·312·58/210·310·56 |
| 2. | (a6/b4)3·(b2/a3)4·a3 = a18/b12·b8/a12·a3 = IV & V = a3/b4 = I & II |
Exercises:
| 1. | Write the number 123456 as a sum of powers of 10! | |
| 2. | Simplify 9·54 - 6·54! | |
| 3. | Rewrite as a power the number 1/256! | |
| 4. | Simplify 5.p2.q4.r6.p.8.r2.q2! |