Evaluate \( 202^2_{three} - 112^2_{three}\)

Explanation

\(202^2_{three}\)when converted to base ten \(=(202_3)^2\\

202_3 = 2 \times 3^2 + 0 \times 3^1 + 2\times 3^0 = 18 + 0 + 2\\

=20_{ten}; (202_3)^2 = (20)^2_{ten} = 400\\

112^2_{three}\)when converted to base ten \(= (112_3)^2\\

112_3 = 1 \times 3^2 + 1 \times 3^1 + 2\times 3^0 = 9+3+2=14_{ten}\\

(112_3)^2 = (14)^2_{ten} = 196_{ten}\\

Evaluate \Longrightarrow 400-196 = 204\)

Reconvert to base three

\(\begin{matrix}

3 & 204\\

3 & 69 &R0\\

3 & 22 & R2\\

3 & 7 & R1\\

2 & 2 & R1\\

& 0& R2 \uparrow\\

\end{matrix} \\

=21120_3\)