Which is equivalent to 64 Superscript one-fourth?
The correct answer and explanation is:
Correct Answer:
64^(1/4) = 2.828
Explanation (300 words):
The expression 64^(1/4) means “the fourth root of 64,” or in other words, what number raised to the power of 4 equals 64?
To understand this, recall how exponents and roots work:
- The square root of a number is the number that when squared gives the original number.
- The fourth root is the number that when raised to the power of 4 gives the original number.
So: x4=64x^4 = 64
We try small whole numbers:
- 14=11^4 = 1
- 24=162^4 = 16
- 34=813^4 = 81
So clearly, 64 is between 16 and 81, and we now know: 2<641/4<32 < 64^{1/4} < 3
To solve this more precisely, convert the expression: 641/4=(26)1/4=26/4=23/264^{1/4} = (2^6)^{1/4} = 2^{6/4} = 2^{3/2}
Now simplify: 23/2=23=82^{3/2} = \sqrt{2^3} = \sqrt{8}
And since: 8=4⋅2=22\sqrt{8} = \sqrt{4 \cdot 2} = 2\sqrt{2}
Using decimal approximation:
- 2≈1.414\sqrt{2} \approx 1.414
- 2⋅1.414≈2.8282 \cdot 1.414 \approx 2.828
Therefore: 641/4=23/2=8≈2.82864^{1/4} = 2^{3/2} = \sqrt{8} \approx 2.828
Final Answer:
641/4≈2.828\boxed{64^{1/4} \approx 2.828}
This demonstrates how fractional exponents can be simplified using root and power rules. It is useful in algebra and higher-level math when solving exponential and radical expressions.