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 :
m
Simplify —
2
Equation at the end of step 1 :
n m
(m-((3•—)•(m3)))+((5•—)•m2)
5 2
Step 2 :
Multiplying exponential expressions :
2.1 m1 multiplied by m2 = m(1 + 2) = m3
Equation at the end of step 2 :
n 5m3 (m-((3•—)•(m3)))+——— 5 2Step 3 :
n Simplify — 5
Equation at the end of step 3 :
n 5m3
(m - ((3 • —) • m3)) + ———
5 2
Step 4 :
Equation at the end of step 4 :
3m3n 5m3
(m - ————) + ———
5 2
Step 5 :
Rewriting the whole as an Equivalent Fraction :
5.1 Subtracting a fraction from a whole
Rewrite the whole as a fraction using 5 as the denominator :
m m • 5
m = — = —————
1 5
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 :
5.2 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:
m • 5 - (3m3n) 5m - 3m3n
—————————————— = —————————
5 5
Equation at the end of step 5 :
(5m - 3m3n) 5m3
——————————— + ———
5 2
Step 6 :
Step 7 :
Pulling out like terms :
7.1 Pull out like factors :
5m - 3m3n = -m • (3m2n - 5)
Trying to factor as a Difference of Squares :
7.2 Factoring: 3m2n - 5
Theory : A difference of two perfect squares, A2 - B2 can be factored into (A+B) • (A-B)
Proof : (A+B) • (A-B) =
A2 - AB + BA - B2 =
A2 - AB + AB - B2 =
A2 - B2
Note : AB = BA is the commutative property of multiplication.
Note : - AB + AB equals zero and is therefore eliminated from the expression.
Check : 3 is not a square !!
Ruling : Binomial can not be factored as the
difference of two perfect squares
Calculating the Least Common Multiple :
7.3 Find the Least Common Multiple
The left denominator is : 5
The right denominator is : 2
| Prime Factor | Left Denominator | Right Denominator | L.C.M = Max {Left,Right} |
|---|---|---|---|
| 5 | 1 | 0 | 1 |
| 2 | 0 | 1 | 1 |
| Product of all Prime Factors | 5 | 2 | 10 |
Least Common Multiple:
10
Calculating Multipliers :
7.4 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
Right_M = L.C.M / R_Deno = 5
Making Equivalent Fractions :
7.5 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. -m • (3m2n-5) • 2 —————————————————— = ————————————————— L.C.M 10 R. Mult. • R. Num. 5m3 • 5 —————————————————— = ——————— L.C.M 10
Adding fractions that have a common denominator :
7.6 Adding up the two equivalent fractions
-m • (3m2n-5) • 2 + 5m3 • 5 -6m3n + 25m3 + 10m
——————————————————————————— = ——————————————————
10 10
Step 8 :
Pulling out like terms :
8.1 Pull out like factors :
-6m3n + 25m3 + 10m = -m • (6m2n - 25m2 - 10)
Trying to factor a multi variable polynomial :
8.2 Factoring 6m2n - 25m2 - 10
Try to factor this multi-variable trinomial using trial and error
Factorization fails
Final result :
+m • (6m2n + 25m2 + 10)
———————————————————————
10
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