Math Problem Statement

Simplify the expression: \( \frac{\frac{\frac{1}{z^3}}{\frac{3}{4}z}}{\frac{3}{4} \cdot \frac{1}{z^4}} \). Write your answer using only positive exponents. Assume that all variables are positive real numbers.

Solution

Let's simplify the given expression step-by-step. The expression in the image is:

1z334z341z4\frac{\frac{\frac{1}{z^3}}{\frac{3}{4}z}}{\frac{3}{4} \cdot \frac{1}{z^4}}

Step 1: Simplify the numerator

The numerator is 1z334z\frac{\frac{1}{z^3}}{\frac{3}{4}z}. To simplify:

1z3÷34z=1z3×43z=43zz3=43z4\frac{1}{z^3} \div \frac{3}{4}z = \frac{1}{z^3} \times \frac{4}{3z} = \frac{4}{3z \cdot z^3} = \frac{4}{3z^4}

So, the simplified numerator is 43z4\frac{4}{3z^4}.

Step 2: Simplify the denominator

The denominator is 341z4\frac{3}{4} \cdot \frac{1}{z^4}:

34×1z4=34z4\frac{3}{4} \times \frac{1}{z^4} = \frac{3}{4z^4}

Step 3: Combine numerator and denominator

Now, we have:

43z434z4=43z4×4z43=4×4z43×3z4=16z49z4\frac{\frac{4}{3z^4}}{\frac{3}{4z^4}} = \frac{4}{3z^4} \times \frac{4z^4}{3} = \frac{4 \times 4z^4}{3 \times 3z^4} = \frac{16z^4}{9z^4}

Canceling out z4z^4 in both the numerator and denominator:

16z49z4=169\frac{16z^4}{9z^4} = \frac{16}{9}

Final Answer:

The simplified expression is 169\frac{16}{9}.

Would you like further details or have any questions? Here are five questions to consider:

  1. What rules did we apply when multiplying fractions?
  2. How did we handle the exponents when simplifying z4z^4?
  3. Can this process be applied to similar rational expressions?
  4. What happens if the exponent values were different (e.g., z5z^5 or z2z^2)?
  5. How do negative exponents affect the simplification process?

Tip: When simplifying expressions, cancel common terms only after you've factored or simplified both numerator and denominator.

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Math Problem Analysis

Mathematical Concepts

Fraction Simplification
Algebra
Exponents
Rational Expressions

Formulas

\( a \div b = a \times \frac{1}{b} \)
Exponent Rules: \( z^m \div z^n = z^{m-n} \)
Multiplying fractions: \( \frac{a}{b} \times \frac{c}{d} = \frac{a \times c}{b \times d} \)

Theorems

Laws of Exponents
Simplification of Rational Expressions

Suitable Grade Level

Grades 10-12