Math Problem Statement

Simplify the expression \( \frac{y^{-1} - (y+3)^{-1}}{3} \) by performing the indicated operations.

Solution

Let's solve the given mathematical expression step by step.

The expression is:

y1(y+3)13\frac{y^{-1} - (y+3)^{-1}}{3}

Step 1: Simplify the numerator

The numerator is y1(y+3)1y^{-1} - (y+3)^{-1}. Recall that x1=1xx^{-1} = \frac{1}{x}, so rewrite the terms:

y1=1y,(y+3)1=1y+3.y^{-1} = \frac{1}{y}, \quad (y+3)^{-1} = \frac{1}{y+3}.

Thus, the numerator becomes:

1y1y+3.\frac{1}{y} - \frac{1}{y+3}.

Step 2: Combine the fractions

To combine the fractions, find the least common denominator (LCD), which is y(y+3)y(y+3). Rewrite the terms with the LCD:

1y=y+3y(y+3),1y+3=yy(y+3).\frac{1}{y} = \frac{y+3}{y(y+3)}, \quad \frac{1}{y+3} = \frac{y}{y(y+3)}.

Now subtract the fractions:

1y1y+3=y+3y(y+3)yy(y+3)=(y+3)yy(y+3).\frac{1}{y} - \frac{1}{y+3} = \frac{y+3}{y(y+3)} - \frac{y}{y(y+3)} = \frac{(y+3) - y}{y(y+3)}.

Simplify the numerator:

(y+3)y=3,(y+3) - y = 3,

so the combined fraction becomes:

1y1y+3=3y(y+3).\frac{1}{y} - \frac{1}{y+3} = \frac{3}{y(y+3)}.

Step 3: Divide by 3

The original expression includes dividing the result by 3. Dividing 3y(y+3)\frac{3}{y(y+3)} by 3 is the same as multiplying the denominator by 3:

3y(y+3)3=33y(y+3).\frac{\frac{3}{y(y+3)}}{3} = \frac{3}{3y(y+3)}.

Simplify:

33y(y+3)=1y(y+3).\frac{3}{3y(y+3)} = \frac{1}{y(y+3)}.

Final Answer:

The simplified expression is:

1y(y+3)\boxed{\frac{1}{y(y+3)}}

Would you like additional clarification or further examples?


Relative Questions:

  1. How do you combine fractions with different denominators?
  2. Why is dividing by 3 equivalent to multiplying the denominator by 3?
  3. What happens if y=0y = 0 or y=3y = -3 in this expression?
  4. Can this expression be expanded further for specific values of yy?
  5. How does this concept relate to simplifying rational expressions?

Tip:

Always check for restrictions in the domain when dealing with fractions to avoid dividing by zero!

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Math Problem Analysis

Mathematical Concepts

Simplification of rational expressions
Negative exponents
Fraction operations

Formulas

x^{-1} = \frac{1}{x}
\frac{a}{b} - \frac{c}{d} = \frac{ad - bc}{bd}

Theorems

Basic properties of exponents

Suitable Grade Level

Grades 9-10