Multiply. Simplify your answer.
1. (n + 3/n- 5) x (n^2 - 5n)
A: n^2 + 3n
2. (6xy^2/2x^2y^6) x (6x^4y^4/9x^3)
A: 2
3. (3h^3 - 6h/10g^2) x (4g/g^2 - 2g)
(This is the correct expression. There is no typo.)
A: ?
4. (m^2 + m - 2/m^2 - 2m - 8) x (m^2 -8 + 16/3m -3)
(This the correct equation. There is no typo.)
A: (m-4)/3
Add or subtract. Simplify your answer.
5. (15)/(2p) - (13)/(2p)
A: 2/2p or p?
6. (3m^2)/(4m^5) + (5m^2)/(4m^5)
A: 2m
7. (x^2 + 8x)/ (x - 2)/ (3x + 14)/ (x - 2)
A: (x + 7)
8. (2t)/(4t^2) + (2)/(t)
A: 10t^2/t^2 or 10?
9. (m^2 - m - 2)/ (m^2 + 6m +5) - (2)/(m + 5)
A: ?
10. (4x)/(x-2) + (3x)/ (2 - x)
A: ?
3. (3h^3 - 6h/10g^2) x (4g/g^2 - 2g)
First, we can simplify both fractions separately:
3h^3 - 6h / 10g^2 = 3h(h^2 - 2) / 10g^2
4g / g^2 - 2g = 4g / g(g - 2) = 4 / (g - 2)
Now, we can multiply the fractions:
(3h(h^2 - 2) / 10g^2) x (4 / (g - 2))
To multiply the fractions, we can cancel out common factors:
(3h(h^2 - 2) x 4) / (10g^2 x (g - 2))
= (12h(h^2 - 2)) / (10g^2(g - 2))
= (6h(h^2 - 2)) / (5g^2(g - 2))
Therefore, the simplified expression is (6h(h^2 - 2)) / (5g^2(g - 2)).
9. (m^2 - m - 2)/ (m^2 + 6m +5) - (2)/(m + 5)
First, let's find a common denominator for the two fractions, which is (m^2 + 6m + 5):
(m^2 - m - 2)(m + 5) / (m^2 + 6m + 5) - 2(m^2 + 6m + 5) / (m + 5)
Next, we can combine the fractions:
((m^2 - m - 2)(m + 5) - 2(m^2 + 6m + 5)) / (m^2 + 6m + 5)
Expanding and simplifying the numerator:
(m^3 + 4m^2 + 3m - 10) - (2m^2 + 12m + 10)
= m^3 + 4m^2 + 3m - 10 - 2m^2 - 12m - 10
= m^3 + 4m^2 + 3m - 2m^2 - 12m - 20
= m^3 + 2m^2 - 9m - 20
Therefore, the simplified expression is m^3 + 2m^2 - 9m - 20.
To solve these multiplication and addition/subtraction problems, follow these steps:
Multiplication:
1. For each multiplication problem, use the distributive property to multiply the terms within the parentheses.
2. Simplify the resulting expression by combining like terms and simplifying any fractions or polynomials.
Addition/Subtraction:
1. For each addition/subtraction problem, ensure that the denominators of the fractions are the same. If they aren't, find a common denominator and adjust the fractions accordingly.
2. Add or subtract the numerators of the fractions while keeping the common denominator.
3. Simplify the resulting fraction, if possible, by reducing it to its simplest form or expressing it as a whole number, if applicable.
4. Simplify any algebraic expressions by combining like terms and performing any necessary operations.
Now let's solve the given problems step by step:
1. (n + 3/n - 5) x (n^2 - 5n)
Applying the distributive property, we get:
n(n^2 - 5n) + 3(n^2 - 5n)
Simplifying each term and combining like terms, we get:
n^3 - 5n^2 + 3n^2 - 15n
Simplifying further, we get:
n^3 - 2n^2 - 15n
2. (6xy^2/2x^2y^6) x (6x^4y^4/9x^3)
Dividing out common factors, we get:
(3y^2)/(x^4y^6) x (2x^3y^4)/(3x^3)
Simplifying further, we get:
2/(x*y^2)
3. (3h^3 - 6h/10g^2) x (4g/g^2 - 2g)
Multiplying the numerators and denominators separately, we get:
3h^3 - 6h/10g^2 * 4g/g^2 - 2g
Simplifying by canceling out like terms and combining like terms, we get:
(12h^3 - 24h)/(10g)
4. (m^2 + m - 2/m^2 - 2m - 8) x (m^2 - 8 + 16/3m - 3)
Multiplying the numerators and denominators separately, we get:
(m^2 + m - 2)(m^2 - 8 + 16)/(m^2 - 2m - 8)(3m - 3)
Simplifying by canceling out like terms and combining like terms, we get:
(m - 4)/(3)
5. (15)/(2p) - (13)/(2p)
Since the denominators are the same, we can subtract the numerators directly:
(15 - 13)/(2p) = 2/(2p) = 1/p
6. (3m^2)/(4m^5) + (5m^2)/(4m^5)
Since the denominators are the same, we can add the numerators directly:
(3m^2 + 5m^2)/(4m^5) = 8m^2/(4m^5)
Simplifying further, we get:
2/m^3
7. (x^2 + 8x)/(x - 2)/(3x + 14)/(x - 2)
To divide by a fraction, we can multiply by its reciprocal:
(x^2 + 8x)/(x - 2) * (x - 2)/(3x + 14)
Simplifying further, we get:
(x^2 + 8x)/(3x + 14)
8. (2t)/(4t^2) + (2)/(t)
Since the denominators are different, we need to find a common denominator:
The common denominator is 4t^2. Adjusting the fractions accordingly, we get:
(2t)/(4t^2) + (2t^2)/(4t^2)
Adding the numerators, we get:
(2t + 2t^2)/(4t^2)
Simplifying further, we get:
(2t^2 + 2t)/(4t^2) = 2t(t + 1)/(4t^2)
Simplifying the fraction, we get:
(t + 1)/(2t)
9. (m^2 - m - 2)/(m^2 + 6m + 5) - (2)/(m + 5)
Finding a common denominator and adjusting the fractions accordingly, we get:
(m^2 - m - 2)/(m^2 + 6m + 5) - (2(m^2 + 6m + 5))/(m + 5)(m^2 + 6m + 5)
Simplifying each term, we get:
(m^2 - m - 2 - 2m^2 - 12m - 10)/(m^3 + 7m^2 + 11m + 25)
Simplifying further, we get:
(-m^2 - 13m - 12)/(m^3 + 7m^2 + 11m + 25)
10. (4x)/(x - 2) + (3x)/(2 - x)
Since the denominators are different, we need to find a common denominator:
The common denominator is (x - 2)(2 - x). Adjusting the fractions accordingly, we get:
(4x(2 - x))/(x - 2)(2 - x) + (3x(x - 2))/(x - 2)(2 - x)
Simplifying each term, we get:
(8x - 4x^2)/(x^2 - 4) + (3x^2 - 6x)/(x^2 - 4)
Combining like terms, we get:
(-4x^2 + 8x + 3x^2 - 6x)/(x^2 - 4)
Simplifying further and combining like terms, we get:
(-x^2 + 2x)/(x^2 - 4)