Simlifying Rational Expressions
Reduce to lowest terms, stating the restrictions on the variable. I really don't understand the whole cancelling out the factor thing, and not cancelling out the terms. As I was having lots of trouble with this stuff this is the way I was told do solve an equation.
5a^3-15a^2/30a
5a^2(a-3)/30a
(5)(a)(a)(a-3)/(3)(2)(5)(a)
a(a-3)/6
a^2-3a/6
a cannot equal 0.
^ But even following it that way I still sort of get confused.
These are the ones I really need help on. And could you please just teach me the rules and stuff for Simplifying rational expressions and stating the restrictions. If you could show me a bunch of examples that would be great, thanks.
Questions - -"Stuck" on.
*a^2/a^2+a
*2-3/r-2
*Tried solving this one...
2b^2-18b/b(b-9)^2
2b(b-9)/b(b-9)(b-9)
I cancelled out the (b-9)'s and I cancelled out the b's( the one beside 2 and the one in the denominator, I wasn't sure whether they are considered as terms and if they should not be cancelled out, but it never made sense to not cancel them out so I did)
And I was left over with : 2/b-9
And the restriction part:
*Imagine the equal sign as a nonequal one
b-9 = 0
Added 9 to both sides
b = 9
Meaning b cannot equal nine. Is that the only restriction? Cause that's the only one I solved.
* t^2+4t+4/2t^2+10t+12
t^2+2t+2t+4/2t^2+6t+4t+12
t(t+2)2(t+2)/2t(t+3)4(t+3)
^ And I'm stuck :\
Questions - -"Stuck" on.
*a^2/a^2+a
= a^2/(a(a+1))
= a/(a+1) , a≠ 0
2-3/r-2 ????? , did you mean (2-r)/(r-2) ?
if so, then ...
= -1(r-2)/(r-2)
= -1 , r≠2
2b^2-18b/b(b-9)^2
I will assume you meant: (2b^2-18b)/( b(b-9)^2 )
= 2b(b-9)/((b-9)(b-9) )
= 2b/(b-9) , b ≠ 9
t^2+4t+4/2t^2+10t+12 , I will again assume you meant (t^2+4t+4)/(2t^2+10t+12)
= (t+2)(t+2)/(2 (t+2)(t+3) )
= (t+2)/(2t+6) , t ≠-3
In the last, it looks like you were factoring by decomposition.
I will do the bottom:
2t^2 + 10t + 12
= 2t^2 + 6t + 4t + 12
= 2t(t+3) + 4(2t+3) , you had these as being multiplied rather than added
= (t+3)(2t+4)
= (t+3)(2)(t+2)
= 2(t+3)(t+2)
Simplifying rational expressions involves reducing fractions to their simplest form by canceling out common factors in the numerator and denominator. To do this, you can follow these steps:
1. Factor both the numerator and denominator completely.
2. Identify any common factors between the numerator and denominator.
3. Cancel out these common factors.
4. Write the resulting expression with the canceled factors.
However, it's important to mention that when canceling out factors, you should only cancel out factors that are multiplied together, not terms added or subtracted. Let's go through the examples you provided to illustrate this process.
1. Simplifying a^2/(a^2 + a):
- Factor the numerator and denominator: a^2 = a * a, and a^2 + a = a(a + 1)
- Cancel out the common factor, which is "a" in this case.
- Write the resulting expression: 1/(a + 1)
- There are no restrictions on the variable "a" in this case.
2. Simplifying 2 - 3/(r - 2):
- There are no common factors to cancel out in the numerator and denominator since they are not factorable.
- Keep the expression as it is: 2 - 3/(r - 2)
- There are no restrictions on the variable "r" in this case.
3. Simplifying 2b^2 - 18b/b(b - 9)^2:
- Factor the numerator: 2b^2 - 18b = 2b(b - 9)
- Factor the denominator: b(b - 9)^2
- Cancel out the common factor (b - 9) in the numerator and denominator.
- Write the resulting expression: 2/(b - 9)
- The restriction on the variable "b" is b ≠ 9 because it would result in a division by zero error.
4. Simplifying t^2 + 4t + 4/2t^2 + 10t + 12:
- Factor the numerator: t^2 + 4t + 4 = (t + 2)(t + 2) = (t + 2)^2
- Factor the denominator: 2t^2 + 10t + 12 = 2(t^2 + 5t + 6) = 2(t + 2)(t + 3)
- Cancel out the common factor (t + 2) in the numerator and denominator.
- Write the resulting expression: (t + 2)/(2t + 6)
- Simplify further by dividing both the numerator and denominator by 2: (t + 2)/(t + 3)
- There are no restrictions on the variable "t" in this case.
Remember to always check for any restrictions on the variable after simplifying a rational expression, as division by zero is not defined.