Evaluate: \((0.064)^-(1)/(3)\)

Mathematics

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Answer

2.5\text{2.5}

Step 1: Convert the decimal to a fraction. 0.064=6410000.064 = \frac{64}{1000}

Step 2: Rewrite the expression with the fraction and apply the negative exponent rule, which states that an=1ana^{-n} = \frac{1}{a^n}. (641000)13=(100064)13\left(\frac{64}{1000}\right)^{-\frac{1}{3}} = \left(\frac{1000}{64}\right)^{\frac{1}{3}}

Step 3: Apply the fractional exponent rule, which states that a1n=ana^{\frac{1}{n}} = \sqrt[n]{a}. (100064)13=1000643\left(\frac{1000}{64}\right)^{\frac{1}{3}} = \sqrt[3]{\frac{1000}{64}}

Step 4: Calculate the cube root of the numerator and the denominator. 1000643=10003643=104\sqrt[3]{\frac{1000}{64}} = \frac{\sqrt[3]{1000}}{\sqrt[3]{64}} = \frac{10}{4}

Step 5: Simplify the fraction. 104=52\frac{10}{4} = \frac{5}{2}

Step 6: Convert the fraction to a decimal. 52=2.5\frac{5}{2} = 2.5

The evaluated expression is: 2.5\boxed{2.5}

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