Q. Find the volume of a rectangular prism measuring \( \tfrac{5}{2}\times\tfrac{5}{2}\times\tfrac{5}{2} \).

Answer

Convert the mixed number to an improper fraction: \(2\frac{1}{2}=\tfrac{5}{2}\). Then
\[
\left(\tfrac{5}{2}\right)^3=\tfrac{5^3}{2^3}=\tfrac{125}{8}=15\frac{5}{8}=15.625.
\]
Final answer: \(\tfrac{125}{8}=15.625\).

Detailed Explanation

Write the expression: \(2\frac{1}{2}\times 2\frac{1}{2}\times 2\frac{1}{2}\).
Convert each mixed number to an improper fraction. For \(2\frac{1}{2}\), multiply the whole number by the denominator and add the numerator: \(2\cdot2+1=5\). So
\[
2\frac{1}{2}=\frac{5}{2}.
\]
Replace each mixed number with the improper fraction:
\[
2\frac{1}{2}\times2\frac{1}{2}\times2\frac{1}{2}
=\frac{5}{2}\times\frac{5}{2}\times\frac{5}{2}.
\]
Multiply the numerators and the denominators:
\[
\frac{5}{2}\times\frac{5}{2}\times\frac{5}{2}
=\frac{5\cdot5\cdot5}{2\cdot2\cdot2}
=\frac{125}{8}.
\]
The fraction \(\tfrac{125}{8}\) is already in lowest terms.
Convert the improper fraction \(\tfrac{125}{8}\) to a mixed number by dividing 125 by 8:
\(125\div8=15\) remainder \(5\). Thus
\[
\frac{125}{8}=15\frac{5}{8}.
\]
Final answer: \(15\frac{5}{8}\) (which is \( \tfrac{125}{8} \) or \(15.625\)).

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FAQs

Q1: What is \(2\ \tfrac{1}{2}\times 2\ \tfrac{1}{2}\times 2\ \tfrac{1}{2}\)?

A1: \(2\ \tfrac{1}{2}=\tfrac{5}{2}\). So \(\left(\tfrac{5}{2}\right)^3=\tfrac{125}{8}=15\ \tfrac{5}{8}=15.625\).

Q2: How do I convert \(2\ \tfrac{1}{2}\) to an improper fraction?

A2: Multiply whole by denominator and add numerator: \(2\cdot2+1=5\). So \(2\ \tfrac{1}{2}=\tfrac{5}{2}\).

Q3: How do I multiply mixed numbers like this?

A3: Convert each to improper fractions, multiply numerators and denominators: \(\tfrac{5}{2}\cdot\tfrac{5}{2}\cdot\tfrac{5}{2}=\tfrac{125}{8}\).

Q4: How do I do it with decimals?

A4: Convert: \(2\ \tfrac{1}{2}=2.5\). Then \(2.5\times2.5=6.25\), and \(6.25\times2.5=15.625\).

Q5: How can I write this using exponents?

A5: It's a cube: \(\left(2\ \tfrac{1}{2}\right)^3=\left(\tfrac{5}{2}\right)^3=125/8\).

Q6: Is there a shortcut to simplify before multiplying?

A6: Here all factors are identical, so use power notation: \(\left(\tfrac{5}{2}\right)^3\). No cross-cancellation is needed; compute numerator and denominator powers.

Q7: How do I convert \(\tfrac{125}{8}\) to a mixed number?

A7: Divide: \(125\div8=15\) remainder \(5\). So \(\tfrac{125}{8}=15\ \tfrac{5}{8}\).

Q8: What is the geometric interpretation?

A8: If a cube has side length \(2.5\) units, its volume is \(V=s^3=(2.5)^3=15.625\) cubic units.

Q9: How do I check my answer with a calculator?

A9: Enter \(2.5^3\) or \(2.5\times2.5\times2.5\); you should get \(15.625\). Or enter \((5/2)^3\) to see \(125/8\).

Q10: How do I get the cube root of the result?

A10: Cube root of \(\tfrac{125}{8}\) is \(\sqrt[3]{\tfrac{125}{8}}=\tfrac{5}{2}=2\ \tfrac{1}{2}\).

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