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Simplify. Assume all variables are positive.\newlinew13w43w73\frac{w^{\frac{1}{3}}}{w^{\frac{4}{3}} \cdot w^{\frac{7}{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.\newlinew13w43w73\frac{w^{\frac{1}{3}}}{w^{\frac{4}{3}} \cdot w^{\frac{7}{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. Write Expression and Combine Exponents: Write down the expression and combine the exponents in the denominator using the property of exponents that states when you multiply like bases, you add the exponents.\newlineExpression: w13/(w43w73)w^{\frac{1}{3}}/(w^{\frac{4}{3}} \cdot w^{\frac{7}{3}})\newlineCombine exponents in the denominator: w43+73w^{\frac{4}{3} + \frac{7}{3}}
  2. Combine Exponents in Denominator: Add the exponents in the denominator.\newlinew43+73=w113w^{\frac{4}{3} + \frac{7}{3}} = w^{\frac{11}{3}}\newlineNow the expression is: w13w113\frac{w^{\frac{1}{3}}}{w^{\frac{11}{3}}}
  3. Subtract Exponents in Numerator: Subtract the exponents in the numerator from the exponents in the denominator using the property of exponents that states when you divide like bases, you subtract the exponents.\newlinew13/w113=w13113w^{\frac{1}{3}} / w^{\frac{11}{3}} = w^{\frac{1}{3} - \frac{11}{3}}
  4. Perform Subtraction: Perform the subtraction of the exponents. w13113=w103w^{\frac{1}{3} - \frac{11}{3}} = w^{-\frac{10}{3}}
  5. Convert Negative Exponent: Since we want the exponent to be positive and we're assuming all variables are positive, we can write the negative exponent as a reciprocal.\newlinew103=1w103w^{-\frac{10}{3}} = \frac{1}{w^{\frac{10}{3}}}

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