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Magazine Chapter 6: Problem 40
Simplify. If possible, use a second method or evaluation as a check. $$ \frac{a^{-1}+b^{-1}}{a^{-3}+b^{-3}} $$
Short Answer
Expert verified
\( \frac{a^3b^2}{b^2-ab+a^2} \)
Step by step solution
01
Understand the given expression
We need to simplify the expression \( \frac{a^{-1}+b^{-1}}{a^{-3}+b^{-3}} \). This expression contains negative exponents, which we will handle first.
02
Rewrite negative exponents as positive exponents
Use the property \(x^{-n} = \frac{1}{x^n}\) to rewrite negative exponents: \(a^{-1} = \frac{1}{a}\), \(b^{-1} = \frac{1}{b}\), \(a^{-3} = \frac{1}{a^3}\), \(b^{-3} = \frac{1}{b^3}\). The expression becomes: \( \frac{\frac{1}{a} + \frac{1}{b}}{\frac{1}{a^3} + \frac{1}{b^3}} \).
03
Find a common denominator for the numerator
Combine the terms in the numerator by finding a common denominator: \( \frac{\frac{1}{a} + \frac{1}{b}}{\frac{1}{a^3} + \frac{1}{b^3}} = \frac{\frac{b+a}{ab}}{\frac{1}{a^3} + \frac{1}{b^3}} \).
04
Find a common denominator for the denominator
Combine the terms in the denominator by finding a common denominator: \( \frac{\frac{b+a}{ab}}{\frac{b^3+a^3}{a^3b^3}} \).
05
Simplify the complex fraction
To simplify the complex fraction, multiply the numerator by the reciprocal of the denominator: \( \frac{\frac{b+a}{ab}}{\frac{b^3+a^3}{a^3b^3}} = \frac{b+a}{ab} \times \frac{a^3b^3}{b^3+a^3} = \frac{(b+a) \cdot a^3b^3}{ab(b^3+a^3)} = \frac{(b+a) \cdot a^3b^2}{b^3+a^3} \).
06
Simplify the final expression
Incorporate the fact that \(b^3+a^3 = (b+a)(b^2-ab+a^2)\): \(\frac{(b+a) \cdot a^3b^2}{(b+a)(b^2-ab+a^2)} = \frac{a^3b^2}{b^2 - ab + a^2} \).
07
Verify the results
Perform the steps again or solve by substituting specific numerical values for \(a\) and \(b\) to verify correctness.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
negative exponents
Negative exponents can often make expressions look complex. But they have a simple rule. When you see a negative exponent, you can rewrite it using positive exponents. The rule is:
For example:
- For any nonzero number x and positive integer n,
- \(x^{-n} = \frac{1}{x^n} \).
For example:
- \(a^{-1} = \frac{1}{a} \)
- \(b^{-1} = \frac{1}{b} \)
- \(a^{-3} = \frac{1}{a^3} \)
- \(b^{-3} = \frac{1}{b^3} \)
common denominators
To add or subtract fractions, we must use a common denominator. The common denominator is a shared multiple of the denominators of the fractions involved.
For instance, to combine \( \frac{1}{a} + \frac{1}{b} \):
For instance, to combine \( \frac{1}{a} + \frac{1}{b} \):
- Find a common denominator for \( a \) and \( b \), which is \( ab \).
- Rewrite each fraction using the common denominator. For \( \frac{1}{a} \), you multiply the numerator and denominator by \( b \), resulting in \(\frac{b}{ab} \).
- For \( \frac{1}{b} \), multiply the numerator and denominator by \( a \), resulting in \(\frac{a}{ab} \).
- Combining both, we get: \( \frac{1}{a} + \frac{1}{b} = \frac{b}{ab} + \frac{a}{ab} = \frac{a + b}{ab} \).
factoring polynomials
Factoring polynomials is the process of breaking down a polynomial into simpler parts (factors) that, when multiplied together, give back the original polynomial.Let's consider the polynomial: \(b^3 + a^3 \). It can be factored using the sum of cubes formula.
- The sum of cubes formula is: \( x^3 + y^3 = (x + y)(x^2 - xy + y^2) \).
- For \( b^3 + a^3\), let \( x = b \) and \( y = a \). So, \( b^3 + a^3 \) becomes \( (b + a)(b^2 - ab + a^2) \).
multiply reciprocals
Multiplying by the reciprocal is a method used to simplify complex fractions. A reciprocal of a number or fraction is simply its 'flipped' version. For a fraction:
- \( \frac{a}{b} \), the reciprocal is \( \frac{b}{a} \).
- When you multiply a number by its reciprocal, you always get 1.
- Flip the denominator's fraction part (reciprocal) and multiply: \( \frac{ \frac{ ( b + a ) }{ ab } }{ \frac{ b^3 + a^3 }{ a^3 b^3 } } = \frac{ b+a }{ ab } \times \frac{ a^3 b^3 }{ b^3+a^3 } \).
- This results in: \( \frac{ ( b + a ) \times a^3 b^2}{ b^3 + a^3 } \).
