Question

1.) (1 - x/x+1) : x/xy+y 2.) b/a + a^2-b^2/a^2 * a/a+b

Answer 1

1. $(1-\frac{x}{x+1}):\frac{x}{xy+y}=(\frac{x+1-x}{x+1})\cdot \frac{y(x+1)}{x}=\frac{1}{x+1}\cdot \frac{y(x+1)}{x}=\frac{y}{x}$

2. $\frac{b}{a}+ \frac{a^2-b^2}{a^2}\cdot \frac{a}{a+b}=\frac{b}{a}+\frac{(a-b)(a+b)}{a^2}\cdot \frac{a}{a+b}=\frac{b}{a}+\frac{a-b}{a}=\frac{b}{a}+1-\frac{b}{a}$

$=1$

Помним, что $\frac{1}{\frac{x}{y} } =\frac{y}{x}$

и что $a^2-b^2=(a-b)(a+b)$

Answer 2

Let's simplify each expression one by one:

1. Simplifying (1 - x/x+1) : x/xy+y: First, we need to clarify the expression by using parentheses: (1 - x/(x+1)) : (x/(xy+y))

   To simplify this expression, we'll start by working on each side of the division sign separately.

   Numerator: 1 - x/(x+1) To combine the terms, we need a common denominator, which is (x+1). 1(x+1)/(x+1) - x/(x+1) (x+1 - x)/(x+1) 1/(x+1)

   Denominator: x/(xy+y) We can factor out an x from the denominator: x/(x(y+1))

   Putting the numerator and denominator together: (1/(x+1)) / (x/(x(y+1)))

   To divide by a fraction, we multiply by its reciprocal: (1/(x+1)) * ((x(y+1))/x) (y+1)/(x+1)

   So, the simplified form of (1 - x/x+1) : x/xy+y is (y+1)/(x+1).

2. Simplifying b/a + a^2 - b^2/a^2 * a/a+b: Again, let's clarify the expression by using parentheses: (b/a) + (a^2 - b^2)/(a^2) * (a/(a+b))

   To simplify this expression, we'll start by simplifying each term separately and then combining them.

   Simplifying each term: b/a is already simplified.

   (a^2 - b^2)/(a^2) This is a difference of squares, which can be factored as (a-b)(a+b). (a-b)(a+b)/(a^2)

   a/(a+b) is already simplified.

   Combining the terms: (b/a) + (a-b)(a+b)/(a^2) * (a/(a+b))

   We can cancel out (a+b) from the numerator and denominator of the second term: (b/a) + (a-b) * a/(a^2)

   Multiplying the numerators and denominators: (b/a) + (a^2 - ab)/(a^2)

   Finding a common denominator: [(b * a) + (a^2 - ab)]/a^2

   Simplifying the numerator: (ab + a^2 - ab)/a^2 (a^2)/a^2

   The expression (a^2)/a^2 simplifies to 1, so the final simplified form is: 1

Therefore, the simplified form of b/a + a^2 - b^2/a^2 * a/a+b is 1.