Math Problem Statement

Simplify the expression: (16a/b - b/a) / (1/a - 4/b).

Solution

Let's simplify the expression given in the image step by step. The expression appears to be:

16abba1a4b\frac{\frac{16a}{b} - \frac{b}{a}}{\frac{1}{a} - \frac{4}{b}}

Step 1: Simplify the Numerator

The numerator is 16abba\frac{16a}{b} - \frac{b}{a}.

To subtract these fractions, we need a common denominator. The common denominator between bb and aa is abab. So we rewrite each fraction:

16ab=16a2ab,ba=b2ab\frac{16a}{b} = \frac{16a^2}{ab}, \quad \frac{b}{a} = \frac{b^2}{ab}

Now, subtract them:

16a2abb2ab=16a2b2ab\frac{16a^2}{ab} - \frac{b^2}{ab} = \frac{16a^2 - b^2}{ab}

Step 2: Simplify the Denominator

The denominator is 1a4b\frac{1}{a} - \frac{4}{b}.

Again, we need a common denominator, which is abab. Rewriting each fraction:

1a=bab,4b=4aab\frac{1}{a} = \frac{b}{ab}, \quad \frac{4}{b} = \frac{4a}{ab}

Now, subtract them:

bab4aab=b4aab\frac{b}{ab} - \frac{4a}{ab} = \frac{b - 4a}{ab}

Step 3: Form the Overall Fraction

Now we have the expression:

16a2b2abb4aab\frac{\frac{16a^2 - b^2}{ab}}{\frac{b - 4a}{ab}}

When dividing fractions, we multiply by the reciprocal of the denominator. So we get:

16a2b2ab×abb4a\frac{16a^2 - b^2}{ab} \times \frac{ab}{b - 4a}

The abab terms cancel out, leaving us with:

16a2b2b4a\frac{16a^2 - b^2}{b - 4a}

Step 4: Factor the Numerator

The numerator 16a2b216a^2 - b^2 is a difference of squares, which can be factored as:

16a2b2=(4ab)(4a+b)16a^2 - b^2 = (4a - b)(4a + b)

Final Expression

Now we have:

(4ab)(4a+b)b4a\frac{(4a - b)(4a + b)}{b - 4a}

Notice that b4ab - 4a is the same as (4ab)-(4a - b). So, we can cancel out the (4ab)(4a - b) terms, but we are left with a negative sign:

(4a+b)-(4a + b)

Thus, the simplified expression is:

(4a+b)-(4a + b)

Final Answer:

(4a+b)-(4a + b)

Would you like further details or have any questions?


Here are five questions for further exploration:

  1. What is the importance of finding a common denominator in fraction operations?
  2. How can you identify the difference of squares when simplifying expressions?
  3. Why does canceling terms in a rational expression require factoring first?
  4. What are the conditions for canceling terms in algebraic fractions?
  5. How does the negative sign affect the result when simplifying fractions?

Tip: Always check if terms in the numerator or denominator can be factored, as it often simplifies the process of solving.

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Math Problem Analysis

Mathematical Concepts

Algebra
Rational Expressions
Factoring
Difference of Squares

Formulas

Common denominator in fractions: a/b - c/d = (ad - bc)/bd
Difference of squares: x^2 - y^2 = (x - y)(x + y)

Theorems

Difference of Squares

Suitable Grade Level

Grades 9-12