(2x + 1/(3y)) ^ 2 - (2x - 1/(3y)) Simplify
pls also give explanation
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Adding, subtracting and finding the least common multiple
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(2x+1/3y)^2-(2x-1/3y)^2
This deals with adding, subtracting and finding the least common multiple.
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STEP
1
:
1
Simplify —
3
Equation at the end of step
1
:
1 1
((2x+(—•y))2)-((2x-(—•y))2)
3 3
STEP
2
:
Rewriting the whole as an Equivalent Fraction
2.1 Subtracting a fraction from a whole
Rewrite the whole as a fraction using 3 as the denominator :
2x 2x • 3
2x = —— = ——————
1 3
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 :
2.2 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:
2x • 3 - (y) 6x - y
———————————— = ——————
3 3
Equation at the end of step
2
:
1 (6x-y)
((2x+(—•y))2)-(——————2)
3 3
STEP
3
:
Equation at the end of step
3
:
1 (6x - y)2
((2x + (— • y))2) - —————————
3 32
STEP
4
:
1
Simplify —
3
Equation at the end of step
4
:
1 (6x - y)2
((2x + (— • y))2) - —————————
3 32
STEP
5
:
Rewriting the whole as an Equivalent Fraction :
5.1 Adding a fraction to a whole
Rewrite the whole as a fraction using 3 as the denominator :
2x 2x • 3
2x = —— = ——————
1 3
Adding fractions that have a common denominator :
5.2 Adding up the two equivalent fractions
2x • 3 + y 6x + y
—————————— = ——————
3 3
Equation at the end of step
5
:
(6x + y) (6x - y)2
(————————)2) - —————————
3 32
STEP
6
:
Equation at the end of step
6
:
(6x + y)2 (6x - y)2
————————— - —————————
32 32
STEP
7
:
7.1 Finding a Common Denominator The left 32
The right 32
The product of any two denominators can be used
as a common denominator.
Said product is not necessarily the least common
denominator.
As a matter of fact, whenever the two denominators
have a common factor, their product will be bigger
than the least common denominator.
Anyway, the product is a fine common denominator and
can perfectly be used for
calculating multipliers, as well as for generating
equivalent fractions.
32 • 32 will be used as a common denominator.
Calculating Multipliers :
7.2 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 = 32
Right_M = L.C.M / R_Deno = 32
Making Equivalent Fractions :
7.3 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. (6x+y)2 • 32
—————————————————— = ————————————
Common denominator 32 • 32
R. Mult. • R. Num. (6x-y)2 • 32
—————————————————— = ————————————
Common denominator 32 • 32
Adding fractions that have a common denominator :
7.4 Adding up the two equivalent fractions
(6x+y)2 • 32 - ((6x-y)2 • 32) 216xy
————————————————————————————— = —————
32 • 32 9 • 9
Final result :
216xy
—————
81
Terms and topics
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