1/16 x² - 1/169x difference of two square
Answers
Step-by-step explanation:
Step
1
:
1
Simplify ———
169
Equation at the end of step
1
:
1 1
(—— • (x2)) - ———
16 169
STEP
2
:
1
Simplify ——
16
Equation at the end of step
2
:
1 1
(—— • x2) - ———
16 169
STEP
3
:
Equation at the end of step 3
x2 1
—— - ———
16 169
STEP
4
:
Calculating the Least Common Multiple :
4.1 Find the Least Common Multiple
The left denominator is : 16
The right denominator is : 169
Number of times each prime factor
appears in the factorization of:
Prime
Factor Left
Denominator Right
Denominator L.C.M = Max
{Left,Right}
2 4 0 4
13 0 2 2
Product of all
Prime Factors 16 169 2704
Least Common Multiple:
2704
Calculating Multipliers :
4.2 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 = 169
Right_M = L.C.M / R_Deno = 16
Making Equivalent Fractions :
4.3 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. x2 • 169
—————————————————— = ————————
L.C.M 2704
R. Mult. • R. Num. 16
—————————————————— = ————
L.C.M 2704
Adding fractions that have a common denominator :
4.4 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:
x2 • 169 - (16) 169x2 - 16
——————————————— = ——————————
2704 2704
Trying to factor as a Difference of Squares:
4.5 Factoring: 169x2 - 16
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 : 169 is the square of 13
Check : 16 is the square of 4
Check : x2 is the square of x1
Factorization is : (13x + 4) • (13x - 4)
Final result :
(13x + 4) • (13x - 4)
—————————————————————
2704