To solve the expression [tex]\(\left(2^6\right)^{-\frac{2}{3}}\)[/tex], we'll proceed step-by-step through the rules of exponents.
1. Evaluate the Inner Exponentiation:
- The expression [tex]\(2^6\)[/tex] represents raising 2 to the power of 6.
- Since [tex]\(2^6 = 64\)[/tex], we can rewrite the original expression as:
[tex]\[
\left(64\right)^{-\frac{2}{3}}
\][/tex]
2. Apply the Negative Exponent Rule:
- The negative exponent rule states that [tex]\(a^{-n} = \frac{1}{a^n}\)[/tex].
- Applying this rule, we get:
[tex]\[
(64)^{-\frac{2}{3}} = \frac{1}{(64)^{\frac{2}{3}}}
\][/tex]
3. Simplify the Fractional Exponent:
- The fractional exponent [tex]\(\frac{2}{3}\)[/tex] indicates a combination of a root and a power:
- The denominator (3) indicates a cube root.
- The numerator (2) indicates squaring.
- Therefore, [tex]\((64)^{\frac{2}{3}}\)[/tex] can be interpreted as the cube root of [tex]\(64\)[/tex] raised to the power of 2:
[tex]\[
(64)^{\frac{2}{3}} = \left(\sqrt[3]{64}\right)^2
\][/tex]
4. Evaluate the Cube Root and Square:
- The cube root of [tex]\(64\)[/tex] is 4, since [tex]\(4^3 = 64\)[/tex].
- Squaring 4, we get [tex]\(4^2 = 16\)[/tex].
Thus, [tex]\((64)^{\frac{2}{3}} = 16\)[/tex].
5. Combine the Results:
- We already had that:
[tex]\[
(64)^{-\frac{2}{3}} = \frac{1}{(64)^{\frac{2}{3}}}
\][/tex]
- Substituting the simplified value:
[tex]\[
(64)^{-\frac{2}{3}} = \frac{1}{16}
\][/tex]
Thus, the final result of the expression [tex]\(\left(2^6\right)^{-\frac{2}{3}}\)[/tex] is:
[tex]\[
\frac{1}{16} \text{ or } 0.0625
\][/tex]
To summarize:
- The intermediate exponent is [tex]\(-4.0\)[/tex].
- The final value of the expression is [tex]\(0.0625\)[/tex].
