Question

question
simplify the following expression to simplest form using only positive exponents.
\\(\left(16x^{28}y^{-12}\
ight)^{\frac{3}{4}}\\)

Explanation:

Step1: Apply exponent rule \((ab)^n = a^n b^n\)

We can distribute the exponent \(\frac{3}{4}\) to each factor inside the parentheses. So we have:
\(16^{\frac{3}{4}} \cdot (x^{28})^{\frac{3}{4}} \cdot (y^{-12})^{\frac{3}{4}}\)

Step2: Simplify \(16^{\frac{3}{4}}\)

Recall that \(16 = 2^4\), so we can rewrite \(16^{\frac{3}{4}}\) as \((2^4)^{\frac{3}{4}}\). Using the exponent rule \((a^m)^n=a^{mn}\), we get \(2^{4\times\frac{3}{4}} = 2^3=8\).

Step3: Simplify \((x^{28})^{\frac{3}{4}}\)

Using the exponent rule \((a^m)^n = a^{mn}\), we multiply the exponents: \(28\times\frac{3}{4}=\frac{84}{4} = 21\). So \((x^{28})^{\frac{3}{4}}=x^{21}\).

Step4: Simplify \((y^{-12})^{\frac{3}{4}}\)

Using the exponent rule \((a^m)^n=a^{mn}\), we multiply the exponents: \(- 12\times\frac{3}{4}=-9\). So \((y^{-12})^{\frac{3}{4}}=y^{-9}\). But we need positive exponents, and recall that \(a^{-n}=\frac{1}{a^n}\), so \(y^{-9}=\frac{1}{y^{9}}\).

Step5: Combine the results

Now we combine the simplified terms from Step2, Step3, and Step4:
\(16^{\frac{3}{4}} \cdot (x^{28})^{\frac{3}{4}} \cdot (y^{-12})^{\frac{3}{4}}=8\times x^{21}\times\frac{1}{y^{9}}=\frac{8x^{21}}{y^{9}}\)

Answer:

\(\frac{8x^{21}}{y^{9}}\)