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Simplify the following expression to simplest form using only positive exponents. (64x^(-24)y^(12))^((4)/(3)) Answer:

Simplify the following expression to simplest form using only positive exponents.\newline(64x−24y12)43 \left(64 x^{-24} y^{12}\right)^{\frac{4}{3}} \newlineAnswer:

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Q. Simplify the following expression to simplest form using only positive exponents.\newline(64x−24y12)43 \left(64 x^{-24} y^{12}\right)^{\frac{4}{3}} \newlineAnswer:
  1. Apply Power to Factors: Apply the power to each factor inside the parentheses.\newlineWe have the expression (64x(−24)y(12))(43)(64x^{(-24)}y^{(12)})^{(\frac{4}{3})}. To simplify, we will apply the exponent (43)(\frac{4}{3}) to each factor inside the parentheses separately.
  2. Simplify Base 6464: Simplify the numerical base 6464 raised to the power of (4/3)(4/3). 6464 is 44 raised to the power of 33 (434^3), so we can rewrite 6464 as 434^3 and then apply the exponent (4/3)(4/3) to it. (43)(4)/(3)=43∗(4/3)=44=256(4^3)^{(4)/(3)} = 4^{3*(4/3)} = 4^4 = 256
  3. Simplify Variable xx: Simplify the variable xx raised to the power of (−24)(-24) and then to the power of (4/3)(4/3). We apply the exponent (4/3)(4/3) to x(−24)x^{(-24)} by multiplying the exponents. (x(−24))(4/3)=x((−24)∗(4/3))=x(−32)(x^{(-24)})^{(4/3)} = x^{((-24)*(4/3))} = x^{(-32)} Since we want only positive exponents, we can write x(−32)x^{(-32)} as 1/x321/x^{32}.
  4. Simplify Variable yy: Simplify the variable yy raised to the power of 1212 and then to the power of (4/3)(4/3). We apply the exponent (4/3)(4/3) to y12y^{12} by multiplying the exponents. (y12)(4/3)=y12∗(4/3)=y16(y^{12})^{(4/3)} = y^{12*(4/3)} = y^{16}
  5. Combine Results: Combine the results from Steps 22, 33, and 44.\newlineWe now combine the simplified terms to get the final expression.\newline256×(1/x32)×y16256 \times (1/x^{32}) \times y^{16}
  6. Final Expression: Write the final expression in simplest form.\newlineThe final expression is already in simplest form with only positive exponents.\newline256⋅y16/x32256 \cdot y^{16} / x^{32}

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