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Problem 42

Simplify the following expression: $$1-[1 /(2-1 / 3)]$$

Expert verified
The simplified expression is: $$\dfrac{-2}{3}$$.
See the step by step solution

Step 1: Simplify the Inner Fraction

We first deal with the inner fraction which is $$\dfrac{1}{3}$$. Since we need to find $$1 \div \dfrac{1}{3}$$, we can rewrite it as $$1 \times \dfrac{3}{1}$$ because division is the same as multiplying by the reciprocal of the fraction. The expression becomes: $$1 - \left[1 \div \left(2 - \dfrac{1}{3}\right)\right] = 1 - \left[1 \times \dfrac{3}{1}\right]$$.

Step 2: Simplify the Expression Inside the Bracket

Now we have to deal with the expression inside the bracket: $$2 - \dfrac{1}{3}$$. To do this, we first find a common denominator for the two fractions. In this case, the common denominator is 3, so we rewrite the expression as: $$\dfrac{6}{3} - \dfrac{1}{3}$$. Subtract the fractions to get: $$\dfrac{5}{3}$$. The expression becomes: $$1 - \left[\dfrac{5}{3}\right]$$.

Step 3: Simplify the Entire Expression

Now we need to simplify the entire expression. We can rewrite it as a single fraction with a common denominator. In this case, the common denominator is 3, so the expression becomes: $$\dfrac{3}{3} - \dfrac{5}{3}$$. Subtract the fractions to get: $$\dfrac{-2}{3}$$. The simplified expression is: $$\dfrac{-2}{3}$$.

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