The simplest form of (64/729) ^-1/6 is -2/33/24/33/4
The Correct Answer and Explanation is:
To simplify the expression: (64729)−16\left(\frac{64}{729}\right)^{-\frac{1}{6}}
✅ Step 1: Apply the negative exponent rule
a−n=1ana^{-n} = \frac{1}{a^n}
So: (64729)−16=(72964)16\left(\frac{64}{729}\right)^{-\frac{1}{6}} = \left(\frac{729}{64}\right)^{\frac{1}{6}}
✅ Step 2: Prime factorization of 729 and 64
- 64=2664 = 2^6
- 729=36729 = 3^6
So: (72964)16=(3626)16=36×1626×16=32\left(\frac{729}{64}\right)^{\frac{1}{6}} = \left(\frac{3^6}{2^6}\right)^{\frac{1}{6}} = \frac{3^{6 \times \frac{1}{6}}}{2^{6 \times \frac{1}{6}}} = \frac{3}{2}
✅ Final Simplified Answer:
(64729)−16=32\left(\frac{64}{729}\right)^{-\frac{1}{6}} = \frac{3}{2}
🔍 Explanation
The expression (64729)−1/6\left(\frac{64}{729}\right)^{-1/6} involves two concepts: negative exponents and rational (fractional) exponents.
First, recall the negative exponent rule, which tells us that: a−n=1ana^{-n} = \frac{1}{a^n}
So, the negative exponent simply inverts the fraction. Thus, (64729)−1/6\left(\frac{64}{729}\right)^{-1/6} becomes (72964)1/6\left(\frac{729}{64}\right)^{1/6}.
Next, we simplify the new base 72964\frac{729}{64}. To do this, factor both numbers into their prime components:
- 64=2664 = 2^6: this is because 2×2×2×2×2×2=642 \times 2 \times 2 \times 2 \times 2 \times 2 = 64.
- 729=36729 = 3^6: because 3×3×3×3×3×3=7293 \times 3 \times 3 \times 3 \times 3 \times 3 = 729.
Therefore: (72964)1/6=(3626)1/6\left(\frac{729}{64}\right)^{1/6} = \left(\frac{3^6}{2^6}\right)^{1/6}
Using the power of a power rule (am)n=amn(a^m)^n = a^{mn}, apply the exponent: 36×1626×16=3121=32\frac{3^{6 \times \frac{1}{6}}}{2^{6 \times \frac{1}{6}}} = \frac{3^1}{2^1} = \frac{3}{2}
So the simplified result is 32\frac{3}{2}.
This result is a rational number and represents the sixth root of the reciprocal of 64729\frac{64}{729}. Despite the initial complexity, recognizing perfect sixth powers and using exponent rules allows for quick simplification. The incorrect answers like “-2/33/24/33/4” appear to be typographical errors or miscalculations. The correct and simplest form is: 32\boxed{\frac{3}{2}}
