Question

Write the following numerals in the indicated base without multiplying out the powers. a. 3•5⁴ +3•5² in base five b. 2• 12⁵ +8•12³ + 12 in base twelve

Answer 1

a. 3•5⁴ + 3•5² in base five is 2333 base five.

b. 2•12⁵ + 8•12³ + 12 in base twelve is 2C0 base twelve.

Explanation:

To convert
`\(3 \cdot 5^4 + 3 \cdot 5^2\)`into base five, recognize that
`\(5^4 = 625\)`and
`\(5^2 = 25\).`Simply substituting these values into the equation, you get
`\(3 \cdot 625 + 3 \cdot 25\),`which equals
`\(1875 + 75 = 1950\`). Now, convert this decimal value into base five, resulting in 2333 base five.

For
`\(2 \cdot 12^5 + 8 \cdot 12^3 + 12\)` in base twelve, note that
`\(12^5 = 248832\)`and
`\(12^3 = 1728\).` Substituting these values into the expression gives
`\(2 \cdot 248832 + 8 \cdot 1728 + 12 = 497664 + 13824 + 12 = 511500\).` This decimal value converted to base twelve becomes 2C0 base twelve.

By recognizing the place values in the respective bases and converting the expressions into decimal form first, the subsequent conversion into the given bases is accomplished.