Math Problem Statement
Solution
The given expression is:
Step 1: Simplify the exponents
- , and .
- Similarly, and .
So the expression becomes:
Step 2: Combine terms in the numerator
For the numerator:
Thus, the numerator is:
Step 3: Combine terms in the denominator
The denominator is:
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:
Step 4: Combine the full expression
Now, divide the numerator by the denominator:
This simplifies by multiplying by the reciprocal of the denominator:
Cancel out common factors:
Final Answer:
The simplified expression is:
Would you like further clarification or details on any step?
5 Related Questions:
- How do you handle negative exponents in general?
- Can you apply this method to other rational expressions involving exponents?
- What happens if in the final simplified expression?
- How does factoring help in simplifying rational expressions?
- Can this expression be simplified further if specific values for and 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
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