Question

simplify the following expression to simplest form using only positive exponents.
\\((64x^{-6}y^{-24})^{-\frac{5}{3}}\\)

Explanation:

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

We can distribute the exponent \(-\frac{5}{3}\) to each factor inside the parentheses:
\(64^{-\frac{5}{3}} \cdot (x^{-6})^{-\frac{5}{3}} \cdot (y^{-24})^{-\frac{5}{3}}\)

Step2: Simplify \(64^{-\frac{5}{3}}\)

First, recall that \(64 = 4^3 = 2^6\). Let's use \(64 = 4^3\) for simplicity.
Using the exponent rule \((a^m)^n = a^{mn}\) and \(a^{-n}=\frac{1}{a^n}\), we have:
\(64^{-\frac{5}{3}}=(4^3)^{-\frac{5}{3}} = 4^{3\times(-\frac{5}{3})}=4^{-5}=\frac{1}{4^5}=\frac{1}{1024}\) Wait, actually, let's use \(64 = 2^6\) or \(64 = 4^3\) correctly. Wait, \(64 = 2^6\), so \(64^{\frac{1}{3}} = 4\) (since \(4^3 = 64\)). So \(64^{-\frac{5}{3}}=(64^{\frac{1}{3}})^{-5}=4^{-5}=\frac{1}{4^5}=\frac{1}{1024}\)? Wait, no, wait, let's do it step by step. Wait, actually, \(a^{-\frac{m}{n}}=\frac{1}{a^{\frac{m}{n}}}=(\frac{1}{a})^{\frac{m}{n}}\), and \(a^{\frac{m}{n}}=\sqrt[n]{a^m}\). So \(64^{\frac{1}{3}}=\sqrt[3]{64}=4\), so \(64^{-\frac{5}{3}}=(64^{\frac{1}{3}})^{-5}=4^{-5}=\frac{1}{4^5}=\frac{1}{1024}\)? Wait, no, that's not right. Wait, actually, let's correct that. Wait, the exponent is \(-\frac{5}{3}\), so:

\(64^{-\frac{5}{3}}=\frac{1}{64^{\frac{5}{3}}}=\frac{1}{(\sqrt[3]{64})^5}=\frac{1}{4^5}=\frac{1}{1024}\)? Wait, no, wait, \(64^{\frac{5}{3}}=(\sqrt[3]{64})^5 = 4^5 = 1024\), so \(64^{-\frac{5}{3}}=\frac{1}{1024}\). Wait, but maybe I made a mistake here. Wait, actually, let's check again. Wait, \(64 = 2^6\), so \(64^{\frac{1}{3}} = 2^{6\times\frac{1}{3}} = 2^2 = 4\), yes, so \(64^{\frac{5}{3}}=(2^6)^{\frac{5}{3}}=2^{10}=1024\), so \(64^{-\frac{5}{3}}=\frac{1}{1024}\)? Wait, no, wait, the original expression is \((64x^{-6}y^{-24})^{-\frac{5}{3}}\), so when we apply the exponent, it's \(64^{-\frac{5}{3}}\), but let's do it with positive exponents from the start. Wait, maybe I messed up the sign. Wait, the exponent is \(-\frac{5}{3}\), so:

\((64x^{-6}y^{-24})^{-\frac{5}{3}} = 64^{-\frac{5}{3}} \cdot x^{(-6)\times(-\frac{5}{3})} \cdot y^{(-24)\times(-\frac{5}{3})}\)

Now, let's compute each term:

For \(64^{-\frac{5}{3}}\):

\(64 = 4^3\), so \(64^{\frac{1}{3}} = 4\), so \(64^{-\frac{5}{3}} = (64^{\frac{1}{3}})^{-5} = 4^{-5} = \frac{1}{4^5} = \frac{1}{1024}\)? Wait, no, wait, actually, \(a^{-n} = \frac{1}{a^n}\), so \(64^{-\frac{5}{3}} = \frac{1}{64^{\frac{5}{3}}}\). And \(64^{\frac{5}{3}} = (\sqrt[3]{64})^5 = 4^5 = 1024\), so that's correct.

Now, for \((x^{-6})^{-\frac{5}{3}}\):

Using the exponent rule \((a^m)^n = a^{mn}\), we have:

\((x^{-6})^{-\frac{5}{3}} = x^{(-6)\times(-\frac{5}{3})} = x^{\frac{30}{3}} = x^{10}\)

For \((y^{-24})^{-\frac{5}{3}}\):

Similarly, \((y^{-24})^{-\frac{5}{3}} = y^{(-24)\times(-\frac{5}{3})} = y^{\frac{120}{3}} = y^{40}\)

Now, putting it all together:

\(64^{-\frac{5}{3}} \cdot x^{10} \cdot y^{40} = \frac{1}{1024} \cdot x^{10} \cdot y^{40}\)? Wait, no, wait, that can't be right. Wait, I think I made a mistake in the sign of the exponent for 64. Wait, the original exponent is \(-\frac{5}{3}\), so when we apply it to 64, which is \(2^6\) or \(4^3\), let's re-express 64 as \(2^6\):

\(64^{-\frac{5}{3}} = (2^6)^{-\frac{5}{3}} = 2^{6\times(-\frac{5}{3})} = 2^{-10} = \frac{1}{2^{10}} = \frac{1}{1024}\). Wait, that's the same as before. But wait, maybe I messed up the initial step. Wait, the expression is \((64x^{-6}y^{-24})^{-\frac{5}{3}}\). Let's use the exponent rule \((a^m)^n = a^{mn}\) correctly.

Wait, another way: \((ab)^n = a^n b^n\), so:

\((64x^{-6}y^{-24})^{-\frac{5}{3}} = 64^{-\frac{5}{3}} \cdot (x^{-6})^{-\frac{5}{3}} \cdot (y^{-24})^{-\frac{5}{3}}\)

Now, \(64 = 4^3\), so \(64^{\frac{1}{3}} = 4\), so \(64^{-\frac{5}{3}} = (4^3)^{-\frac{5}{3}} = 4^{-5} = \frac{1}{4^5} = \frac{1}{1024}\). Correct.

\((x^{-6})^{-\frac{5}{3}} = x^{(-6)\times(-\frac{5}{3})} = x^{10}\). Correct.

\((y^{-24})^{-\frac{5}{3}} = y^{(-24)\times(-\frac{5}{3})} = y^{40}\). Correct.

So multiplying them together: \(\frac{1}{1024}x^{10}y^{40}\)? Wait, but that seems odd. Wait, maybe I made a mistake in the sign of the exponent for 64. Wait, no, the exponent is \(-\frac{5}{3}\), so it's correct. Wait, but let's check with a different approach. Let's take the reciprocal first.

\((64x^{-6}y^{-24})^{-\frac{5}{3}} = \frac{1}{(64x^{-6}y^{-24})^{\frac{5}{3}}}\)

Now, compute \((64x^{-6}y^{-24})^{\frac{5}{3}}\):

\(64^{\frac{5}{3}} \cdot (x^{-6})^{\frac{5}{3}} \cdot (y^{-24})^{\frac{5}{3}}\)

\(64^{\frac{5}{3}} = (\sqrt[3]{64})^5 = 4^5 = 1024\)

\((x^{-6})^{\frac{5}{3}} = x^{-10}\)

\((y^{-24})^{\frac{5}{3}} = y^{-40}\)

So \((64x^{-6}y^{-24})^{\frac{5}{3}} = 1024x^{-10}y^{-40}\)

Therefore, \(\frac{1}{(64x^{-6}y^{-24})^{\frac{5}{3}}} = \frac{1}{1024x^{-10}y^{-40}} = \frac{x^{10}y^{40}}{1024}\)

Which is the same as \(\frac{1}{1024}x^{10}y^{40}\). Wait, but that seems correct. Wait, but let's check with \(64 = 2^6\):

\(64^{\frac{5}{3}} = (2^6)^{\frac{5}{3}} = 2^{10} = 1024\), correct.

\((x^{-6})^{\frac{5}{3}} = x^{-10}\), correct.

\((y^{-24})^{\frac{5}{3}} = y^{-40}\), correct.

So the reciprocal is \(\frac{x^{10}y^{40}}{1024}\), which is the same as \(\frac{1}{1024}x^{10}y^{40}\). Wait, but maybe I made a mistake in the exponent sign for 64. Wait, no, the original exponent is \(-\frac{5}{3}\), so taking the reciprocal of the entire expression raised to \(\frac{5}{3}\) is correct.

Wait, let's verify with a simpler example. Suppose we have \((a^m)^{-n} = a^{-mn} = \frac{1}{a^{mn}}\), which is the same as \((\frac{1}{a^m})^n = \frac{1}{a^{mn}}\). So that's correct.

Alternatively, let's use \(64 = 2^6\):

\((64x^{-6}y^{-24})^{-\frac{5}{3}} = (2^6 x^{-6} y^{-24})^{-\frac{5}{3}} = 2^{6\times(-\frac{5}{3})} x^{-6\times(-\frac{5}{3})} y^{-24\times(-\frac{5}{3})} = 2^{-10} x^{10} y^{40} = \frac{x^{10} y^{40}}{2^{10}} = \frac{x^{10} y^{40}}{1024}\)

Yes, that's correct. So the simplified form is \(\frac{x^{10} y^{40}}{1024}\) or \(\frac{1}{1024}x^{10}y^{40}\).

Answer:

\(\boxed{\dfrac{x^{10} y^{40}}{1024}}\)