Solution - Adding, subtracting and finding the least common multiple
Other Ways to Solve
Adding, subtracting and finding the least common multipleStep by Step Solution
Step 1 :
c
Simplify ——————
4 + 4d
Step 2 :
Pulling out like terms :
2.1 Pull out like factors :
4 + 4d = 4 • (d + 1)
Equation at the end of step 2 :
c c
(cd+(———————•16))-(2•———————)
(cd-8d) 4•(d+1)
Step 3 :
Equation at the end of step 3 :
c c
(cd+(———————•16))-———————
(cd-8d) 2•(d+1)
Step 4 :
c
Simplify ———————
cd - 8d
Step 5 :
Pulling out like terms :
5.1 Pull out like factors :
cd - 8d = d • (c - 8)
Equation at the end of step 5 :
c c
(cd+(———————•16))-———————
d•(c-8) 2•(d+1)
Step 6 :
Equation at the end of step 6 :
16c c
(cd + ———————————) - ———————————
d • (c - 8) 2 • (d + 1)
Step 7 :
Rewriting the whole as an Equivalent Fraction :
7.1 Adding a fraction to a whole
Rewrite the whole as a fraction using d • (c-8) as the denominator :
cd cd • d • (c - 8)
cd = —— = ————————————————
1 d • (c - 8)
Equivalent fraction : The fraction thus generated looks different but has the same value as the whole
Common denominator : The equivalent fraction and the other fraction involved in the calculation share the same denominator
Adding fractions that have a common denominator :
7.2 Adding up the two equivalent fractions
Add the two equivalent fractions which now have a common denominator
Combine the numerators together, put the sum or difference over the common denominator then reduce to lowest terms if possible:
cd • d • (c-8) + 16c c2d2 - 8cd2 + 16c
———————————————————— = —————————————————
d • (c-8) d • (c - 8)
Equation at the end of step 7 :
(c2d2 - 8cd2 + 16c) c
——————————————————— - ———————————
d • (c - 8) 2 • (d + 1)
Step 8 :
Step 9 :
Pulling out like terms :
9.1 Pull out like factors :
c2d2 - 8cd2 + 16c = c • (cd2 - 8d2 + 16)
Trying to factor a multi variable polynomial :
9.2 Factoring cd2 - 8d2 + 16
Try to factor this multi-variable trinomial using trial and error
Factorization fails
Calculating the Least Common Multiple :
9.3 Find the Least Common Multiple
The left denominator is : d • (c - 8)
The right denominator is : 2 • (d + 1)
| Prime Factor | Left Denominator | Right Denominator | L.C.M = Max {Left,Right} |
|---|---|---|---|
| 2 | 0 | 1 | 1 |
| Product of all Prime Factors | 1 | 2 | 2 |
| Algebraic Factor | Left Denominator | Right Denominator | L.C.M = Max {Left,Right} |
|---|---|---|---|
| d | 1 | 0 | 1 |
| c - 8 | 1 | 0 | 1 |
| d + 1 | 0 | 1 | 1 |
Least Common Multiple:
2d • (c - 8) • (d + 1)
Calculating Multipliers :
9.4 Calculate multipliers for the two fractions
Denote the Least Common Multiple by L.C.M
Denote the Left Multiplier by Left_M
Denote the Right Multiplier by Right_M
Denote the Left Deniminator by L_Deno
Denote the Right Multiplier by R_Deno
Left_M = L.C.M / L_Deno = 2•(d + 1)
Right_M = L.C.M / R_Deno = d•(c - 8)
Making Equivalent Fractions :
9.5 Rewrite the two fractions into equivalent fractions
Two fractions are called equivalent if they have the same numeric value.
For example : 1/2 and 2/4 are equivalent, y/(y+1)2 and (y2+y)/(y+1)3 are equivalent as well.
To calculate equivalent fraction , multiply the Numerator of each fraction, by its respective Multiplier.
L. Mult. • L. Num. c • (cd2-8d2+16) • 2 • (d+1) —————————————————— = ———————————————————————————— L.C.M 2d • (c-8) • (d+1) R. Mult. • R. Num. c • d • (c-8) —————————————————— = —————————————————— L.C.M 2d • (c-8) • (d+1)
Adding fractions that have a common denominator :
9.6 Adding up the two equivalent fractions
c • (cd2-8d2+16) • 2 • (d+1) - (c • d • (c-8)) 2c2d3 + 2c2d2 - c2d - 16cd3 - 16cd2 + 40cd + 32c
—————————————————————————————————————————————— = ————————————————————————————————————————————————
2d • (c-8) • (d+1) 2d • (c - 8) • (d + 1)
Step 10 :
Pulling out like terms :
10.1 Pull out like factors :
2c2d3 + 2c2d2 - c2d - 16cd3 - 16cd2 + 40cd + 32c =
c • (2cd3 + 2cd2 + cd + 16d3 + 16d2 + 40d + 32)
Final result :
c • (2cd3 + 2cd2 + cd + 16d3 + 16d2 + 40d + 32)
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