SIMPLIFYING COMPLEX RATIONAL EXPRESSIONS

Example 1 :

Solution  :

By taking the least common multiple of numerator and denominator, we get

So, the answer is (x + 4)/2.

Example 2 :

Solution  :

By taking the least common multiple of the denominator, we get

=  (1 - m)/m² - 1²

=  -(-1 + m)/(m + 1)(m - 1)

=  - 1/(m + 1)

So, the answer is - 1/(m + 1).

Example 3 :

Solution  :

By taking the least common multiple of numerator and denominator, we get

So, the answer is (x + 4)/4

Example 4 :

Solution  :

By using cross multiplication, we get

So, the answer is (-s)/[r(r + s)]

Example 5 :

Solution  :

By using cross multiplication, we get

So, the answer is 1/(x - 1)

Example 6 :

Solution  :

By cross multiplication, we get

So, the answer is a/(a + 1)

Example 7 :

Solution  :

By taking the least common multiple of numerator and denominator, we get

So, the answer is (x + 1)/(x - 3)

Example 8 :

Solution  :

By cross multiplication, we get

So, the answer is 1/(2a²).

Example 9 :

[5x/(x + 2)] ÷ [10/(x - 2)]

Solution :

[5x/(x + 2)] ÷ [10/(x - 2)]

= [5x/(x + 2)] ⋅ [(x - 2)/10]

= 5x(x - 2)/10(x + 2)

= x(x - 2)/2(x + 2)

Example 10 :

[(x/y²) + (1/y)] ÷ [(y/x²) + (1/x)]

Solution :

= [(x/y²) + (1/y)] ÷ [(y/x²) + (1/x)]

= [(x + y) / y²] ÷ [(y +  x)/x²]

= [(x + y) / y²] ⋅ [x² / (y +  x)]

= x² / y²

Example 11 :

[8x/(x - 4)] ÷ [3/(x + 4)]

Solution :

= [8x/(x - 4)] ÷ [3/(x + 4)]

= [8x/(x - 4)] ⋅ [(x + 4)/3]

Simplification is not possible here, then multiply the numerator by numerator multiply the denominator by denominator.

= 8x(x + 4)/3(x - 4)

Example 12 :

[(b/a²) + (1/a)] ÷ [(a/b²) + (1/b)]

Solution :

= [(b/a²) + (1/a)] ÷ [(a/b²) + (1/b)]

= [(b + a) / a²] ÷ [(a + b)/b²]

= [(a + b) / a²] ⋅ [b² / (a + b)]

= b² / a²

Example 13 :

[(4x² - y²)/xy] ÷ [(2/y) - (1/x)]

Solution :

= [(4x² - y²)/xy] ÷ [(2/y) - (1/x)]

= [(2x)² - y²)/xy] ÷ [(2x - y)/xy]

= [(2x + y) (2x - y)/xy] ÷ [(2x - y)/xy]

= [(2x + y) (2x - y)/xy] ⋅ [xy /(2x - y)]

= 2x + y

Example 14 :

(a^-1 + 1)/(a^-1 - 1)

Solution :

(a^-1 + 1)/(a^-1 - 1)

= (1/a + 1) / (1/a - 1)

= (1 + a)/a / (1 - a)/a

= [(1 + a)/a] ⋅ [a/(1 - a)]

= (1 + a)/(1 - a)

Example 15 :

[3x^-1 + (2y)^-1] / x^-2

Solution :

= [3x^-1 + (2y)^-1] / x^-2

= [3/x + (1/2y)] / (1/x²)

= [(6y + x)/2xy] / (1/x²)

(x + 6y) / 2xy  ⋅ (x² / 1)

= x(x + 6y)/2y

Example 16 :

[5x^-2 - 3y^-1] / (x^-1 +  y^-1)

Solution :

= [5x^-2 - 3y^-1] / (x^-1 +  y^-1)

= [5/x² - 3/y] / ((1/x) + (1/y)]

= [(5y - 3x²)/(x²y)] / [(x + y)/xy]

= [(5y - 3x²)/(x²y)] ⋅ [xy / (x + y)]

= (5y - 3x²)/x (x + y)

Example 17 :

[a^-1 + 2a^-2] / [2a^-1 + (2a)^-1]

Solution :

= [a^-1 + 2a^-2] / [2a^-1 + (2a)^-1]

= [1/a + 2/a²] / [2/a + 1/2a]

= [(a + 2)/a²] / [(4 + 1)/2a]

= [(a + 2)/a²] / [5/2a]

= [(a + 2)/a²] ⋅ [2a/5]

= 2(a + 2)/5a

Example 18 :

[5x^-1 - 2y^-1] / [25x^-2 - 4y^-2]

Solution :

= [5x^-1 - 2y^-1] / [25x^-2 - 4y^-2]

= [5/x - 2/y] / [(25/x²) - (4/y²)]

= [(5y - 2x)/xy] / [(25y² - 4x²)/x² y²]

= [(5y - 2x)/xy] / [(5y)² - (2x)²]/x² y²]

= [(5y - 2x)/xy] / [(5y + 2x)(5y - 2x)/x² y²]

= [(5y - 2x)/xy] ⋅ [x² y²/(5y + 2x)(5y - 2x)]

= xy/(5y + 2x)

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