Remaining amount = 800 × (½)^2.5 = 800 × (1/√(2³)) = 800 × (1/√8) ≈ 800 × 0.35355 ≈ <<800*0.35355=282.84>>282.84 g - Decision Point
Understanding the Calculation: Remaining Amount = 800 × (½)^2.5 ≈ 282.84 grams
Understanding the Calculation: Remaining Amount = 800 × (½)^2.5 ≈ 282.84 grams
When working with exponential expressions involving fractions and roots, precise calculations are key—especially in scientific, financial, or engineering contexts. One such calculation that frequently appears is:
Remaining amount = 800 × (½)^2.5
Understanding the Context
At first glance, this expression may seem complex, but breaking it down step-by-step reveals its clarity and practical significance.
Decoding the Expression
We begin with:
800 × (½)^2.5
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Key Insights
This combines a base fraction, raised to a non-integer exponent, multiplied by 800. Let’s simplify step by step.
Step 1: Rewrite (½)^2.5 using rational exponents and roots
The exponent 2.5 can be expressed as a fraction:
2.5 = 2 + 0.5 = 5/2
So,
(½)^2.5 = (½)^(5/2)
This means we take the ½ raised to the 5th power, then take the square root (since exponent over 2 means root):
(½)^(5/2) = (√½)^5 = (1/√2)^5
But for easier calculation, recall:
½ = (1/2) = (1/√8) × √8 / √8 = better simplified as √(1/4) = 1/√4 = 1/2
But closest in simplified radical form:
1/√(2³), because 2³ = 8, and √8 = √(2³) = 2√2 ≈ 2.828
So:
800 × (½)^2.5 = 800 × (1/√8)
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Step 2: Evaluate (1/√8) numerically
We know:
√8 = √(4 × 2) = √4 × √2 = 2√2 ≈ 2 × 1.4142 = 2.8284
So:
1/√8 ≈ 1 / 2.8284 ≈ 0.35355
Step 3: Multiply by 800
Now compute:
800 × 0.35355 ≈ 282.84 grams
Practical Application & Summary
This calculation — 800 × (½)^2.5 ≈ 282.84 — shows how fractional exponents simplify into radical forms involving square roots. In real-world scenarios, such expressions appear in:
