Arithmetic · real student question

Evaluate (1/4) squared divided by (1/4) squared, plus (1/2) squared, plus 4/3, minus (4/3) cubed divided by (4/3) squared, minus 1/2.

Question

Evaluate (14)2:(14)2+(12)2+43−(43)3:(43)2−(12)1.\left(\frac14\right)^2:\left(\frac14\right)^2+\left(\frac12\right)^2+\frac43-\left(\frac43\right)^3:\left(\frac43\right)^2-\left(\frac12\right)^1.

Step-by-step solution

  1. Identify the structure. There are no brackets, so every power and division must be resolved before any addition or subtraction. The expression is a sum of five completed terms.

  2. Evaluate the self-quotient. (14)2:(14)2=1,\left(\frac14\right)^2:\left(\frac14\right)^2 = 1, since any nonzero quantity divided by itself is 11 - there is no need to compute 116\tfrac1{16}.

  3. Evaluate the remaining powers. (12)2=14\left(\tfrac12\right)^2 = \tfrac14 and (12)1=12\left(\tfrac12\right)^1 = \tfrac12.

  4. Use the quotient rule on the base 4/3. (43)3:(43)2=(43)3−2=43.\left(\frac43\right)^3:\left(\frac43\right)^2 = \left(\frac43\right)^{3-2} = \frac43.

  5. Substitute and spot the cancellation. The expression becomes 1+14+43−43−12,1+\frac14+\frac43-\frac43-\frac12, and the two 43\tfrac43 terms cancel exactly.

  6. Add what is left over a common denominator of 4. 44+14−24=34.\frac44+\frac14-\frac24 = \frac34. Note the trap: the 43\tfrac43 terms only cancel because the division is performed before the subtraction.

Answer

34\frac{3}{4}

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