Arithmetic · real student question

Evaluate [(1/2) squared divided by (1/2) cubed] times [3/2 minus (4/9) squared divided by (1 - 5/9) squared].

Question

Evaluate [(12)2:(12)3]⋅[32−(49)2:(1−59)2].\left[\left(\frac12\right)^2 : \left(\frac12\right)^3\right]\cdot\left[\frac32-\left(\frac49\right)^2:\left(1-\frac59\right)^2\right].

Step-by-step solution

  1. Plan the order. Two independent brackets are multiplied at the end, so evaluate each one completely first. Inside each bracket the usual precedence applies: brackets, then powers, then division, then subtraction.

  2. Simplify the first bracket with the quotient rule. Same base, so subtract exponents: (12)2:(12)3=(12)2−3=(12)−1=2.\left(\frac12\right)^2:\left(\frac12\right)^3 = \left(\frac12\right)^{2-3} = \left(\frac12\right)^{-1} = 2. A negative exponent flips the fraction - no need to compute 14\tfrac14 and 18\tfrac18 separately.

  3. Simplify the inner subtraction of the second bracket. 1−59=99−59=49.1-\frac59 = \frac99-\frac59 = \frac49. This is the step that makes the whole problem collapse.

  4. Do the division inside the second bracket. Both squares are now identical: (49)2:(49)2=1681:1681=1.\left(\frac49\right)^2:\left(\frac49\right)^2 = \frac{16}{81}:\frac{16}{81} = 1.

  5. Finish the second bracket. 32−1=12.\frac32-1 = \frac12.

  6. Multiply the two brackets. 2⋅12=1.2\cdot\frac12 = 1. The designed cancellations - a negative exponent giving 22 and a self-quotient giving 11 - make the final answer exactly 11.

Answer

11

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