IDNLearn.com provides a seamless experience for finding accurate answers. Our experts provide accurate and detailed responses to help you navigate any topic or issue with confidence.
Sagot :
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].
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].
Thank you for being part of this discussion. Keep exploring, asking questions, and sharing your insights with the community. Together, we can find the best solutions. IDNLearn.com is your reliable source for accurate answers. Thank you for visiting, and we hope to assist you again.