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Simplify the following expression to simplest form using only positive exponents. (32x^(5)y^(-40))^(-(3)/(5)) Answer:

Simplify the following expression to simplest form using only positive exponents.\newline(32x5y−40)−35 \left(32 x^{5} y^{-40}\right)^{-\frac{3}{5}} \newlineAnswer:

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Q. Simplify the following expression to simplest form using only positive exponents.\newline(32x5y−40)−35 \left(32 x^{5} y^{-40}\right)^{-\frac{3}{5}} \newlineAnswer:
  1. Apply negative exponent rule: Apply the negative exponent rule to the entire expression.\newlineThe negative exponent rule states that a−n=1ana^{-n} = \frac{1}{a^n}. We will apply this rule to the entire expression.\newline(32x5y−40)−35=1(32x5y−40)35(32x^{5}y^{-40})^{-\frac{3}{5}} = \frac{1}{(32x^{5}y^{-40})^{\frac{3}{5}}}
  2. Distribute exponents to factors: Distribute the exponent to each factor inside the parentheses.\newlineWhen raising a product to an exponent, we raise each factor to that exponent.\newline1(32x5y−40)35=1(3235)(x5∗(35))(y−40∗(35))\frac{1}{(32x^{5}y^{-40})^{\frac{3}{5}}} = \frac{1}{(32^{\frac{3}{5}})(x^{5*(\frac{3}{5})})(y^{-40*(\frac{3}{5})})}
  3. Simplify each factor: Simplify each factor separately.\newlineWe will simplify the exponents for each factor.\newline323532^{\frac{3}{5}} is the fifth root of 3232 cubed. The fifth root of 3232 is 22, so (3235)=23=8(32^{\frac{3}{5}}) = 2^3 = 8.\newlinex5∗(35)x^{5*(\frac{3}{5})} simplifies to x3x^{3}.\newliney−40∗(35)y^{-40*(\frac{3}{5})} simplifies to y−24y^{-24}.\newlineSo, 1((3235)(x5∗(35))(y−40∗(35)))=1((23)(x3)(y−24))\frac{1}{((32^{\frac{3}{5}})(x^{5*(\frac{3}{5})})(y^{-40*(\frac{3}{5})}))} = \frac{1}{((2^3)(x^3)(y^{-24}))}
  4. Convert negative exponents: Convert negative exponents to positive exponents.\newlineWe will move the factor with a negative exponent from the denominator to the numerator to make the exponent positive.\newline1(23)(x3)(y−24)=y24(23)(x3)\frac{1}{(2^3)(x^3)(y^{-24})} = \frac{y^{24}}{(2^3)(x^3)}
  5. Write final simplified expression: Write the final simplified expression.\newlineThe final expression with only positive exponents is:\newliney248x3\frac{y^{24}}{8x^3}

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