Simplify the following expression to simplest form using only positive exponents. (32x^5y^-35)^-(3)/(5) Answer Attemptiout of 2 square

To simplify the expression \((32x^{5}y^{-35})^{-\frac{3}{5}}\), we will apply the rules of exponents step by step.<br /><br />First, let's rewrite the expression inside the parentheses with positive exponents:<br />\[ 32x^5y^{-35} \]<br /><br />We can rewrite \(y^{-35}\) as \(\frac{1}{y^{35}}\):<br />\[ 32x^5 \cdot \frac{1}{y^{35}} = \frac{32x^5}{y^{35}} \]<br /><br />Now, we need to apply the exponent \(-\frac{3}{5}\) to this fraction:<br />\[ \left( \frac{32x^5}{y^{35}} \right)^{-\frac{3}{5}} \]<br /><br />When raising a fraction to a power, we raise both the numerator and the denominator to that power:<br />\[ \frac{(32x^5)^{-\frac{3}{5}}}{(y^{35})^{-\frac{3}{5}}} \]<br /><br />Next, apply the exponent \(-\frac{3}{5}\) to both the numerator and the denominator separately.<br /><br />For the numerator:<br />\[ (32x^5)^{-\frac{3}{5}} \]<br />\[ 32^{-\frac{3}{5}} \cdot (x^5)^{-\frac{3}{5}} \]<br /><br />For \(32^{-\frac{3}{5}}\), recall that \(32 = 2^5\):<br />\[ (2^5)^{-\frac{3}{5}} = 2^{-3} = \frac{1}{2^3} = \frac{1}{8} \]<br /><br />For \((x^5)^{-\frac{3}{5}}\):<br />\[ x^{5 \cdot -\frac{3}{5}} = x^{-3} = \frac{1}{x^3} \]<br /><br />So, the numerator becomes:<br />\[ \frac{1}{8} \cdot \frac{1}{x^3} = \frac{1}{8x^3} \]<br /><br />For the denominator:<br />\[ (y^{35})^{-\frac{3}{5}} \]<br />\[ y^{35 \cdot -\frac{3}{5}} = y^{-21} = \frac{1}{y^{21}} \]<br /><br />So, the denominator becomes:<br />\[ \frac{1}{y^{21}} \]<br /><br />Now, combine the simplified numerator and denominator:<br />\[ \frac{\frac{1}{8x^3}}{\frac{1}{y^{21}}} = \frac{1}{8x^3} \cdot y^{21} = \frac{y^{21}}{8x^3} \]<br /><br />Thus, the simplified form of the expression is:<br />\[ \boxed{\frac{y^{21}}{8x^3}} \]