Could anyone help me to slove these problems.
1. (x/(x^2-100))9x(x+10)/(x+3))
2. (x/x^2-100))/(9x(x+10)/(9x(x+10)/(x+3))
3. What is the least common multiple of the expressions: 2x,x^3,(x+1),(x+1)^2,(x-9)
4.(2y/x^2)+(7/x(x-3))
5.(2y/x^2)-(7/x(x-3))
Of course! I'll guide you through solving these problems step by step.
1. (x/(x^2-100)) * (9x(x+10)/(x+3))
To simplify this expression, we can start by factoring the denominator (x^2-100) as a difference of squares: (x-10)(x+10).
Next, we can simplify the expression further by canceling out common factors. Notice that we have an x in both the numerator and denominator, so we can cancel them out:
(x/(x-10)(x+10)) * (9(x+10)/(x+3))
Now, multiply the numerators together and the denominators together:
= (x * 9(x+10)) / ((x-10)(x+10)(x+3))
Simplifying further, we get:
= (9x(x+10)) / ((x-10)(x+10)(x+3))
So, the simplified form of the expression is (9x(x+10)) / ((x-10)(x+10)(x+3)).
2. (x/(x^2-100)) / (9x(x+10)/(9x(x+10)/(x+3)))
To simplify this expression, we can start by dividing the numerator by the denominator. When dividing fractions, you can invert the second fraction and multiply:
= (x/(x^2-100)) * ((9x(x+10)/(9x(x+10)/(x+3)))^-1)
Now, let's invert the second fraction:
= (x/(x^2-100)) * ((x+3)/(9x(x+10)/(9x(x+10))))
Next, multiply the numerators together and the denominators together:
= (x(x+3)) / [(x^2-100) * (9x(x+10) / 9x(x+10))]
Notice that the (x+10) terms in the numerator and denominator will cancel out:
= (x(x+3)) / (x^2-100)
Finally, expand the numerator:
= (x^2+3x) / (x^2-100)
So, the simplified form of the expression is (x^2+3x) / (x^2-100).
3. To find the least common multiple (LCM) of the expressions 2x, x^3, (x+1), (x+1)^2, (x-9), we need to factor each expression completely.
2x: Prime factorization is 2 * x.
x^3: Prime factorization is x * x * x.
(x+1): This is already factored completely.
(x+1)^2: This expression is already factored completely.
(x-9): This is already factored completely.
Now, look for the highest power of each prime factor. The LCM will be the product of these highest powers. In this case, the LCM is:
2 * x * x * x * (x+1)^2 * (x-9)
So, the least common multiple of the expressions 2x, x^3, (x+1), (x+1)^2, (x-9) is 2x^4(x+1)^2(x-9).
4. (2y/x^2) + (7/x(x-3))
To add these fractions, we need a common denominator. In this case, the common denominator is x^2(x-3). To get each fraction to have this common denominator, we need to multiply the numerators and denominators accordingly:
= (2y * x(x-3) + 7 * x^2) / (x^2 * (x-3))
Multiplying the numerators:
= (2xy(x-3) + 7x^2) / (x^2 * (x-3))
Now, combine like terms in the numerator:
= (2xy(x-3) + 7x^2) / (x^2 * (x-3))
So, the simplified form of the expression is (2xy(x-3) + 7x^2) / (x^2 * (x-3)).
5. (2y/x^2) - (7/x(x-3))
To subtract these fractions, we also need a common denominator. Following the same steps as in the previous problem, we can multiply the numerators and denominators accordingly:
= (2y * x(x-3) - 7 * x^2) / (x^2 * (x-3))
Multiplying the numerators:
= (2xy(x-3) - 7x^2) / (x^2 * (x-3))
Now, simplify or expand the numerator as needed:
= (2xy(x-3) - 7x^2) / (x^2 * (x-3))
So, the simplified form of the expression is (2xy(x-3) - 7x^2) / (x^2 * (x-3)).