Question

Help! What is the least common denominator of 4/x^2y+5/y^2?

Answer 1

4/x2y-5/xy2
Final result :

-y • (5xy - 4)
——————————————
x2

Step by step solution :
Step 1 :

Raise y to the 2nd power

Exponentiaion :

Equation at the end of step 1 :

4 5
(————•y)-(—•y2)
(x2) x

Step 2 :

Raise x to the 2nd power

Exponentiaion :

Equation at the end of step 2 :

4 5y2
(—— • y) - ———
x2 x

Step 3 :

4y 5y2
Simplify —— - ———
x2 x

Calculating the Least Common Multiple :

3.1 Find the Least Common Multiple

The left denominator is : x2

The right denominator is : x
Number of times each Algebraic Factor
appears in the factorization of: Algebraic
Factor Left
Denominator Right
Denominator L.C.M = Max
{Left,Right}
x 2 1 2


Least Common Multiple:
x2
Calculating Multipliers :

3.2 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 = 1

Right_M = L.C.M / R_Deno = x

Making Equivalent Fractions :

3.3 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. 4y
—————————————————— = ——
L.C.M x2

R. Mult. • R. Num. 5y2 • x
—————————————————— = ———————
L.C.M x2

Adding fractions that have a common denominator :

3.4 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:

4y - (5y2 • x) 4y - 5xy2
—————————————— = —————————
x2 x2

Pulling out like terms :

3.5 Pull out like factors :

4y - 5xy2 = -y • (5xy - 4)
Final result :

-y • (5xy - 4)
——————————————
x2

Answer 2

Thank you. Could you help me with one more? A/5+a/x+5?

Answer 3

a/5+a/x+5
Final result :

ax + 5a + 25x
—————————————
5x

Step by step solution :
Step 1 :

a a
Simplify — + —
5 x

Calculating the Least Common Multiple :

1.1 Find the Least Common Multiple

The left denominator is : 5

The right denominator is : x
Number of times each prime factor
appears in the factorization of: Prime
Factor Left
Denominator Right
Denominator L.C.M = Max
{Left,Right}
5 1 0 1
Product of all
Prime Factors 5 1 5

Number of times each Algebraic Factor
appears in the factorization of: Algebraic
Factor Left
Denominator Right
Denominator L.C.M = Max
{Left,Right}
x 0 1 1


Least Common Multiple:
5x
Calculating Multipliers :

1.2 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 = x

Right_M = L.C.M / R_Deno = 5

Making Equivalent Fractions :

1.3 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. a • x
—————————————————— = —————
L.C.M 5x

R. Mult. • R. Num. a • 5
—————————————————— = —————
L.C.M 5x

Adding fractions that have a common denominator :

1.4 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:

a • x + a • 5 ax + 5a
————————————— = ———————
5x 5x

Equation at the end of step 1 :

(ax + 5a)
————————— + 5
5x

Step 2 :

ax+5a
Simplify ————— + 5
5x

Rewriting the whole as an Equivalent Fraction :

2.1 Adding a whole to a fraction

Rewrite the whole as a fraction using 5x as the denominator :

5 5 • 5x
5 = — = ——————
1 5x

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
Pulling out like terms :

2.2 Pull out like factors :

ax + 5a = a • (x + 5)

Adding fractions that have a common denominator :

2.3 Adding up the two equivalent fractions

a • (x+5) + 5 • 5x ax + 5a + 25x
—————————————————— = —————————————
5x 5x

Final result :

ax + 5a + 25x
—————————————
5x
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