Write a rational expression in x which simplifies to x + 3 x − 10 and where x ∉ { 6 , 10 } .

We know that the values of a variable that are not allowed in the denominator are called the restrictions on the variable. Since x ∉ 6 , 10 so denominator should be either x - 6 x - 10 or multiple of x - 6 x - 10 . The one possible expression can be x + 3 x - 6 x - 6 x + 10 which simplifies to x + 3 x - 10 .

Simplify the rational expression 25 p 55 k × 66 p 5 k . State any restrictions on the variable.

We have, 25 p 55 k × 66 p 5 k Factorise the numerator and denominator completely. = 5 × 5 × p 5 × 11 × k × 6 × 11 × p 5 × k Now simplifying the expression. = p k × 6 × p k = 6 × p 2 k 2 . Here, the restriction is k ≠ 0 .

Simplify the rational expression 3 p 3 c 4 5 c 5 ÷ 24 p 5 c 10 c 3 . State any restrictions on the variable.

We have, 3 p 3 c 4 5 c 5 ÷ 24 p 5 c 10 c 3 Factorize the numerator and denominator completely. = 3 × p × p × p × c × c × c × c 5 × c × c × c × c × c ÷ 2 × 2 × 2 × 3 × p × p × p × p × p × c 2 × 5 × c × c × c To divide by a fraction, multiply by its reciprocal. = 3 × p × p × p × c × c × c × c 5 × c × c × c × c × c × 2 × 5 × c × c × c 2 × 2 × 2 × 3 × p × p × p × p × p × c Now simplifying the expression. = c 2 × 2 × p × p = c 4 p 2 Here, the restriction is c , p ≠ 0

Simplify the rational expression 5 x + 10 3 x − 9 × 6 x − 18 4 x + 8 . State any restrictions on the variable.

We have, 5 x + 10 3 x − 9 × 6 x − 18 4 x + 8 Factorize the numerator and denominator completely. = 5 x + 2 3 x − 3 × 6 x − 3 4 x + 2 Now simplifying the expression. = 5 2 Here, the restriction is x ∉ { − 2 , 3 }

Simplify the rational expression 6 x − 12 y x + 3 y ÷ 3 2 x + 9 y . State any restrictions on the variable.

We have, 6 x − 12 y x + 3 y ÷ 3 2 x + 9 y Factorize the numerator and denominator completely. = 6 x − 2 y x + 3 y ÷ 3 2 x + 9 y To divide by a fraction, multiply by its reciprocal. = 6 ( x − 2 y ) x + 3 y × 2 x + 9 y 3 Now simplifying the expression. = 2 ( x − 2 y ) 2 x + 9 y x + 3 y Here, the restriction is x ∉ − 9 2 y , − 3 y

Simplify the rational expression x 2 − x − 2 x 2 − 2 x + 1 × x − 1 x + 1 . State any restrictions on the variable.

We have, x 2 − x − 2 x 2 − 2 x + 1 × x − 1 x + 1 = x 2 − 2 x + x − 2 x 2 - x - x + 1 × x − 1 x + 1 Factorize the numerator and denominator completely. = x x − 2 + 1 x − 2 x x - 1 - 1 x - 1 × x − 1 x + 1 Now simplifying the expression. = x + 1 x − 2 x - 1 x - 1 × x − 1 x + 1 = x − 2 x - 1 Here, the restriction is x ∉ { − 1 , 1 }

Simplify the rational expression a 3 + a b 2 a 3 − 2 a 2 b ÷ a 2 + b 2 ( a − 2 b ) 2 . State any restrictions on the variable.

We have, a 3 + a b 2 a 3 − 2 a 2 b ÷ a 2 + b 2 ( a − 2 b ) 2 Factorize the numerator and denominator completely. = a a 2 + b 2 a 2 a − 2 b ÷ a 2 + b 2 ( a − 2 b ) 2 To divide by a fraction, multiply by its reciprocal. = a a 2 + b 2 a 2 a − 2 b × ( a − 2 b ) 2 a 2 + b 2 Now simplifying the expression. = ( a − 2 b ) a Here, the restriction is a ∉ { 0 , 2 b } .

Simplify the rational expression x 2 − 3 x − 10 x 2 + x − 6 × x 2 + 2 x − 3 x 2 + x − 2 . State any restrictions on the variable.

We have, x 2 − 3 x − 10 x 2 + x − 6 × x 2 + 2 x − 3 x 2 + x − 2 = x 2 − 5 x + 2 x − 10 x 2 + 3 x - 2 x − 6 × x 2 + 3 x - x − 3 x 2 + 2 x - x − 2 Factorize the numerator and denominator completely. = x x − 5 + 2 x − 5 x x + 3 - 2 x + 3 × x x + 3 - 1 x + 3 x x + 2 - 1 x + 2 = x + 2 x − 5 x - 2 x + 3 × x - 1 x + 3 x - 1 x + 2 Now simplifying the expression. = x − 5 x - 2 Here, the restriction is x ∉ { − 3 , − 2 , 1 , 2 } .