Math Problem Statement

Simplify the expression (3y^-1 + 4x^-1) / (9y^-2 - 16x^-2).

Solution

The given expression is:

3y1+4x19y216x2\frac{3y^{-1} + 4x^{-1}}{9y^{-2} - 16x^{-2}}

Step 1: Simplify the exponents

  • y1=1yy^{-1} = \frac{1}{y}, and x1=1xx^{-1} = \frac{1}{x}.
  • Similarly, y2=1y2y^{-2} = \frac{1}{y^2} and x2=1x2x^{-2} = \frac{1}{x^2}.

So the expression becomes:

3y+4x9y216x2\frac{\frac{3}{y} + \frac{4}{x}}{\frac{9}{y^2} - \frac{16}{x^2}}

Step 2: Combine terms in the numerator

For the numerator:

3y+4x=3x+4yxy\frac{3}{y} + \frac{4}{x} = \frac{3x + 4y}{xy}

Thus, the numerator is:

3x+4yxy\frac{3x + 4y}{xy}

Step 3: Combine terms in the denominator

The denominator is:

9y216x2\frac{9}{y^2} - \frac{16}{x^2}

This is a difference of squares, which can be factored as:

\frac{9}{y^2} - \frac{16}{x^2} = \frac{(3/y - 4/x)(3/y + 4/x)}

Thus, the denominator becomes:

(3x4y)(3x+4y)x2y2\frac{(3x - 4y)(3x + 4y)}{x^2 y^2}

Step 4: Combine the full expression

Now, divide the numerator by the denominator:

3x+4yxy(3x4y)(3x+4y)x2y2\frac{\frac{3x + 4y}{xy}}{\frac{(3x - 4y)(3x + 4y)}{x^2 y^2}}

This simplifies by multiplying by the reciprocal of the denominator:

3x+4yxy×x2y2(3x4y)(3x+4y)=xyx2y2(3x4y)(3x+4y)xy\frac{3x + 4y}{xy} \times \frac{x^2 y^2}{(3x - 4y)(3x + 4y)} = \frac{xy \cdot x^2 y^2}{(3x - 4y)(3x + 4y) \cdot xy}

Cancel out common factors:

13x4y\frac{1}{3x - 4y}

Final Answer:

The simplified expression is:

13x4y\boxed{\frac{1}{3x - 4y}}

Would you like further clarification or details on any step?

5 Related Questions:

  1. How do you handle negative exponents in general?
  2. Can you apply this method to other rational expressions involving exponents?
  3. What happens if x=yx = y in the final simplified expression?
  4. How does factoring help in simplifying rational expressions?
  5. Can this expression be simplified further if specific values for xx and yy are known?

Tip:

When simplifying fractions with exponents, always try to convert negative exponents into fractions to make the algebra easier.

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Math Problem Analysis

Mathematical Concepts

Algebra
Negative Exponents
Rational Expressions

Formulas

y^-1 = 1/y
x^-1 = 1/x
y^-2 = 1/y^2
x^-2 = 1/x^2
Difference of squares: a^2 - b^2 = (a - b)(a + b)

Theorems

Difference of squares
Exponent rules

Suitable Grade Level

Grades 10-12