Math Problem Statement

Simplify the expression: (a^(-2) - 4) / (b^(-2) - b^(-1) - 2)

Solution

The expression provided in the image is:

a24b2b12\frac{a^{-2} - 4}{b^{-2} - b^{-1} - 2}

Let's simplify this step by step.

Step 1: Simplify the numerator a24a^{-2} - 4

  • The term a2a^{-2} is equivalent to 1a2\frac{1}{a^2}.
  • So, the numerator becomes: 1a24\frac{1}{a^2} - 4

Step 2: Simplify the denominator b2b12b^{-2} - b^{-1} - 2

  • b2b^{-2} is equivalent to 1b2\frac{1}{b^2}.
  • b1b^{-1} is equivalent to 1b\frac{1}{b}.
  • So, the denominator becomes: 1b21b2\frac{1}{b^2} - \frac{1}{b} - 2

Step 3: Write the simplified expression

Now, the expression can be written as: 1a241b21b2\frac{\frac{1}{a^2} - 4}{\frac{1}{b^2} - \frac{1}{b} - 2}

Step 4: Find a common denominator and simplify further

Let's factor the expression further if possible. For simplicity, let's observe if we can factorize the numerator and denominator to find any common terms. However, in this specific form, the expression is in its simplified state without specific values for aa and bb.

Result: 1a241b21b2\frac{\frac{1}{a^2} - 4}{\frac{1}{b^2} - \frac{1}{b} - 2}

This is the simplified expression.

Would you like more details, or do you have any questions? Here are some related questions:

  1. How would you simplify an expression involving negative exponents further if given specific values?
  2. Can you factor the denominator more effectively in this expression?
  3. What happens if a=1a = 1 and b=1b = 1 in the simplified expression?
  4. How would the expression change if b2b1b^{-2} - b^{-1} were the only terms in the denominator?
  5. What are the steps to evaluate this expression for given values of aa and bb?

Tip: When simplifying expressions involving exponents, always remember to handle negative exponents by converting them into fractions.

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Math Problem Analysis

Mathematical Concepts

Algebra
Negative Exponents
Rational Expressions

Formulas

Negative Exponent: a^(-n) = 1/(a^n)
Rational Expressions Simplification

Theorems

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Suitable Grade Level

Grades 9-11