Simplify, stating any restrictions on the variables:
a. 18a^6bc^7d^3/ 12a^2b^4c^7d
b.x^2+5x+6/x^2-2x-15
c. 6y^2-96/2y^2+18y+40
I will assume that in each case you are dividing the entire expression, thus you will need
brackets around it.
e.g.
b) (x^2+5x+6)/(x^2-2x-15)
= (x+2)(x+3)/( (x+3)(x-5) )
= (x+2)/(x-5) , x ≠ -3,5
a) 18a^6bc^7d^3/(12a^2b^4c^7d)
= (18/12)(a^6/a^2)(b/b^4)(c^7/c^7)(d^3/d)
= (3/2)(a^4)(1/b^3)(1)(d^2)
= ..... , a,b,c,d ≠ 0
3rd one is just like the 2nd
Is there an error in the first one? You have ≠-3,5 isn't it suppose to be -2 instead of -3?
@Reiny
a. To simplify 18a^6bc^7d^3/12a^2b^4c^7d, we can simplify the numbers and simplify the variables separately:
First, simplify the numbers: 18/12 = 3/2
Next, simplify the variables:
a^6/a^2 = a^(6-2) = a^4
b/b^4 = 1/b^(4-1) = 1/b^3
c^7/c^7 = 1 (since any non-zero number raised to the power of 0 is 1)
d^3/d = d^(3-1) = d^2
Combining both the simplified numbers and variables, we have: (3/2) * a^4 * (1/b^3) * 1 * d^2
So, the simplified expression is: (3/2) * a^4 * (1/b^3) * d^2
Restrictions: There are no restrictions on the variables a, b, c, and d.
b. To simplify (x^2 + 5x + 6)/(x^2 - 2x - 15), we can factor the numerator and denominator
Factor numerator: (x + 3)(x + 2)
Factor denominator: (x - 5)(x + 3)
Notice that (x + 3) is common to both numerator and denominator, so we can cancel it out:
(x + 3)/(x + 3) = 1
Now we have simplified expression: (x + 2)/(x - 5)
Restrictions: The variable x cannot be equal to -3 or 5 since they would make the denominator zero, which is not allowed in the expression.
c. To simplify (6y^2 - 96)/(2y^2 + 18y + 40), we can factor the numerator and denominator.
Factor numerator: 6(y^2 - 16) = 6(y + 4)(y - 4)
Factor denominator: 2(y^2 + 9y + 20) = 2(y + 4)(y + 5)
Notice that (y + 4) is common to both numerator and denominator, so we can cancel it out:
(6(y - 4))/(2(y + 5))
Simplifying further:
3(y - 4)/(y + 5)
Restrictions: The variable y cannot be equal to -4 or -5 since they would make the denominator zero, which is not allowed in the expression.