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Simplify. Assume all variables are positive.\newlined43d43d43\frac{d^{\frac{4}{3}}}{d^{\frac{4}{3}} \cdot d^{\frac{4}{3}}}\newlineWrite your answer in the form AA or AB\frac{A}{B}, where AA and BB are constants or variable expressions that have no variables in common. All exponents in your answer should be positive.\newline______

Full solution

Q. Simplify. Assume all variables are positive.\newlined43d43d43\frac{d^{\frac{4}{3}}}{d^{\frac{4}{3}} \cdot d^{\frac{4}{3}}}\newlineWrite your answer in the form AA or AB\frac{A}{B}, where AA and BB are constants or variable expressions that have no variables in common. All exponents in your answer should be positive.\newline______
  1. Identify Given Expression: Write down the given expression and identify the properties of exponents to use.\newlineThe given expression is d43/(d43d43)d^{\frac{4}{3}}/(d^{\frac{4}{3}} * d^{\frac{4}{3}}). We will use the properties of exponents to simplify the expression, specifically the quotient of powers property and the product of powers property.
  2. Apply Product Property: Apply the product of powers property to the denominator.\newlineWhen multiplying powers with the same base, we add the exponents. So, we have:\newlined43d43=d43+43=d83d^{\frac{4}{3}} \cdot d^{\frac{4}{3}} = d^{\frac{4}{3} + \frac{4}{3}} = d^{\frac{8}{3}}.
  3. Rewrite with Simplified Denominator: Rewrite the original expression with the simplified denominator.\newlineNow, the expression is:\newlined43/d83d^{\frac{4}{3}} / d^{\frac{8}{3}}.
  4. Apply Quotient Property: Apply the quotient of powers property to simplify the expression.\newlineWhen dividing powers with the same base, we subtract the exponents. So, we have:\newlined43/d83=d4383=d43d^{\frac{4}{3}} / d^{\frac{8}{3}} = d^{\frac{4}{3} - \frac{8}{3}} = d^{-\frac{4}{3}}.
  5. Rewrite with Positive Exponent: Since we want the exponent to be positive, we can rewrite the expression with a positive exponent. d(4/3)d^{(-4/3)} is the same as 1/d(4/3)1/d^{(4/3)}.
  6. Final Answer Form: Write the final answer in the form AA or AB\frac{A}{B}.\newlineThe final answer is in the form AB\frac{A}{B}, where A=1A=1 and B=d43B=d^{\frac{4}{3}}.

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