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Simplify the following expression to simplest form using only positive exponents. (32x^(10)y^(5))^(-(1)/(5)) Answer:

Simplify the following expression to simplest form using only positive exponents.\newline(32x10y5)−15 \left(32 x^{10} y^{5}\right)^{-\frac{1}{5}} \newlineAnswer:\newline

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Full solution

Q. Simplify the following expression to simplest form using only positive exponents.\newline(32x10y5)−15 \left(32 x^{10} y^{5}\right)^{-\frac{1}{5}} \newlineAnswer:\newline
  1. Understand Problem: Understand the problem and apply the negative exponent rule.\newlineThe negative exponent rule states that a−n=1ana^{-n} = \frac{1}{a^n}. We will apply this rule to the entire expression.
  2. Apply Negative Exponent Rule: Apply the negative exponent rule to the expression.\newline(32x10y5)−(1)/(5)=1(32x10y5)1/5(32x^{10}y^{5})^{-(1)/(5)} = \frac{1}{(32x^{10}y^{5})^{1/5}}
  3. Simplify Expression Inside Parentheses: Simplify the expression inside the parentheses by taking the fifth root.\newlineThe fifth root of 3232 is 22, because 25=322^5 = 32.\newlineThe fifth root of x10x^{10} is x10/5=x2x^{10/5} = x^2, because (x2)5=x2⋅5=x10(x^2)^5 = x^{2\cdot5} = x^{10}.\newlineThe fifth root of y5y^{5} is y5/5=y1=yy^{5/5} = y^1 = y, because y1=yy^1 = y.\newline1(32x10y5)1/5=12x2y\frac{1}{(32x^{10}y^{5})^{1/5}} = \frac{1}{2x^2y}
  4. Write Final Simplified Expression: Write the final simplified expression with only positive exponents.\newlineThe final expression is 12x2y\frac{1}{2x^2y}.

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