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# Rational expressions and least common multiple

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Find all numbers for which the rational expression is undefined.
-15/23z

Find all numbers for which the rational expression is undefined.
(t^(3 )-6t)/(t^2-25)

Simplify by removing factor of 1.
(〖80v〗^( 8 ) y^9)/(〖50v〗^(4 ) y^6 ) (The simplified form is)

Simplify by removing factor of 1.
(r^(2 )-16)/(r^(2 )-8r+16)

Simplify by removing factor of 1
(〖21v〗^(2 )-189)/(〖49v〗^(2 )-441)

Multiply and simplify
〖7b〗^(7 )/〖11v〗^4 ∙ 〖121v〗^8/49b

Multiply and simplify

Divide and simplify
f/b^2 ÷ f^2/b^3

Divide and simplify
(x^(2 )-36)/x ÷ (x-6)/(x+7) (Type a fraction. Leave your answer in factored form)

Divide and simplify
(x^(2 )-1)/(49x+49) ÷ (x-1)/7 (Type a fraction)

Divide and simplify
(z^(2 )+4z)/(z^(2 )+2z-8) ÷ z/(z+2) (Type a fraction)

Find the LCM of
〖10x〗^4, 〖50x〗^6

Find the LCM of
q^(2 )-49,q^(2 )+12q+35

Find the LCM of
w^(3 )+ 〖10w〗^(2 )+25w, w^(2 )-10w

〖9y〗^2+45y,〖 3y〗^2+24y+45

6/(8+z) +1/(8+z)

(x^(2 )+13x)/(x^(2 )-8x) +(x^(2 )-6x)/(x^(2 )-8x)

7/〖25z〗^3 +1/〖35z〗^2

6x/(x^(2 )-9) +x/(x-3)

(w+1)/w +w/(w+1)

Subtract and simplify if possible
(z-9)/z-(3z-17)/7z (Simplify your answer. Use integers or fraction for any numbers in the expression)

3/(〖7s〗^(2 )-7s) -7/(7s-7) ((Simplify your answer. Use integers or fraction for any numbers in the expression)

Subtract. Simplify by removing a factor of 1 when possible
12bc/b^(2 -c^2 ) - (b-c)/(b+c)

Subtract. Simplify if possible

Perform the indicated operation and simplify
(x-2)/(x-4)-(x+1)/(x+4)+(x-20)/(x^(2 )-16)

Simplifying the expression
(a²-1)/(a-1)-a

Simplify
((x^2 - 1) / (x - 1)) - 1

https://brainmass.com/math/basic-algebra/rational-expressions-least-common-multiple-210429

## SOLUTION This solution is FREE courtesy of BrainMass!

Please see the attached file for detailed solution.

1. Find all numbers for which the rational expression is undefined.

The denominator cannot be zero, so

Thus, the expression is undefined at 0.

2. Find all numbers for which the rational expression is undefined.

The denominator cannot be zero, so first find out the values that make the denominator 0, then we can exclude those values from the expression:

Thus, the expression is undefined at -5 and 5.

3. Simplify by removing factor of 1.
(The simplified form is)

The property of the exponents we used in above simplification is

4. Simplify by removing factor of 1.

We first factor the numerator using the difference of squares formula and the denominator by perfect square formula, then cancel out the common factors in the numerator and denominator:

5. Simplify by removing factor of 1

First factor the numerator and denominator by extracting out the greatest common factor (GCF) separately , and then cancel out the common factors in the numerator and denominator:

6. Multiply and simplify

7. Multiply and simplify
Factor the trinomials first:
By grouping,
By perfect square formula:

Then cancel out the common factors in the numerator and denominator:

8. Divide and simplify

Flip the numerator and the denominator of the second fraction to convert the division to multiplication, then simplify:

9. Divide and simplify

Difference of squares formula:

10. Divide and simplify
(Type a fraction)

11. Divide and simplify
(Type a fraction)
Factor by extracting out the GCF z:
Factor the trinomial by grouping:

Flip the numerator and the denominator of the second fraction to convert the division to multiplication, then simplify:

12. Find the LCM of
,
The LCM of 10 and 50 is 50
The LCM of x4 and x6 is x6
So the LCM of 10x4 and 50x6 is 50x6.

13. Find the LCM of

Factor the two expressions:

The LCM is the product of all factors:

14. Find the LCM of
+

Then the LCM is

15.

The LCM of 3 and 9 is 9.
So the LCM is

From No. 16 to No. 28, the concepts are:
(a) factor the denominators if possible
(b) find the LCD (least common denominator)
(c) write the fractions using the LCD as denominators
(d) perform the necessary operations (add or subtract the numerators)
(e) simplify the final single fraction if possible

Since they already have a common denominator, we just add the numerator together:

Again, they already have a common denominator, so we just add the numerator together:

The LCM of 25 and 35:
25 = 5 * 5
35 = 5 * 7
So LCM is 5 * 5 * 7 = 175
The LCM of z2 and z3 is z3.
Therefore, the LCD is 175z3.
First write the fractions using the LCD as the denominators, and then simplify:

Factor the denominator in the first fraction:

The second denominator is (x - 3)
So the LCD is (x -3)(x + 3)
Thus

21. Add and simplify if possible

The LCD is w(w +1)
So

22. Subtract and simplify if possible
(Simplify your answer. Use integers or fraction for any numbers in the expression)

23. ((Simplify your answer. Use integers or fraction for any numbers in the expression)

24. Subtract. Simplify by removing a factor of 1 when possible

After factoring out the denominator, there is no common factors in the numerator and the denominator.

25. Subtract. Simplify if possible

26. Perform the indicated operation and simplify

The last denominator can be factored as x2 - 16 = (x - 4)(x + 4)
So the LCD is (x - 4)(x + 4), then

27. Simplifying the expression
(a²-1)/(a-1)-a

28. Simplify
((x^2 - 1) / (x - 1)) - 1

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