Question
Simplify.
(–64)^–4/3
(–64)^–4/3
Answers
GPT-4o mini
To simplify \((-64)^{-4/3}\), we start by applying the negative exponent rule, which states that \(a^{-n} = \frac{1}{a^n}\). Thus, we can rewrite the expression as:
\[
(-64)^{-4/3} = \frac{1}{(-64)^{4/3}}
\]
Next, we need to calculate \((-64)^{4/3}\). This expression can be interpreted as taking the cube root first and then raising the result to the fourth power.
First, we find the cube root of \(-64\):
\[
\sqrt[3]{-64} = -4
\]
Now we raise \(-4\) to the power of \(4\):
\[
(-4)^4 = 256
\]
Putting it all together, we have:
\[
(-64)^{4/3} = 256
\]
Now we substitute this back into the expression:
\[
(-64)^{-4/3} = \frac{1}{(-64)^{4/3}} = \frac{1}{256}
\]
Thus, the simplified form of \((-64)^{-4/3}\) is
\[
\boxed{\frac{1}{256}}
\]
\[
(-64)^{-4/3} = \frac{1}{(-64)^{4/3}}
\]
Next, we need to calculate \((-64)^{4/3}\). This expression can be interpreted as taking the cube root first and then raising the result to the fourth power.
First, we find the cube root of \(-64\):
\[
\sqrt[3]{-64} = -4
\]
Now we raise \(-4\) to the power of \(4\):
\[
(-4)^4 = 256
\]
Putting it all together, we have:
\[
(-64)^{4/3} = 256
\]
Now we substitute this back into the expression:
\[
(-64)^{-4/3} = \frac{1}{(-64)^{4/3}} = \frac{1}{256}
\]
Thus, the simplified form of \((-64)^{-4/3}\) is
\[
\boxed{\frac{1}{256}}
\]