Complete each equation. 31. \( 5^{2} \cdot 5^{\mathrm{m}}=5^{\text {11 }} \) 32. \( m^{\oplus} \cdot m^{-4}=m^{-9} \) 33. \( 2^{(m-2)} \cdot 2^{1} \) 34. \( a^{m} \cdot a^{4}=1 \) 35. \( a^{l} \cdot a^{-2}=a^{l} \) 36. \( x^{3} y^{\text {b }} \cdot x^{5}-y^{2} \)

To solve each equation, we'll apply the laws of exponents and simplify accordingly. 31. To complete the equation \( 5^{2} \cdot 5^{m} = 5^{11} \), we add the exponents: \( 2 + m = 11 \). Therefore, \( m = 11 - 2 = 9 \). 32. In the equation \( m^{\oplus} \cdot m^{-4} = m^{-9} \), we combine the exponents: \( \oplus - 4 = -9 \). Hence, \( \oplus = -9 + 4 = -5 \). 33. For \( 2^{(m-2)} \cdot 2^{1} \), we add the exponents: \( (m - 2) + 1 = m - 1 \). 34. The equation \( a^{m} \cdot a^{4} = 1 \) implies that the exponent must equal zero: \( m + 4 = 0 \). Thus, \( m = -4 \). 35. In \( a^{l} \cdot a^{-2} = a^{l} \), we set the combined exponent equal to the exponent on the right: \( l - 2 = l \). This results in \( 0 = 2 \), which is a contradiction; hence, there are no solutions. 36. For \( x^{3} y^{b} \cdot x^{5} y^{-2} = x^{8} y^{b - 2} \), we add the exponents of \( x \) and combine the exponents of \( y \) directly. Thus, it's equivalent to \( x^{8} y^{b - 2} \). Here's the compiled list of solutions: 31. \( m = 9 \) 32. \( \oplus = -5 \) 33. \( 2^{(m-1)} \) 34. \( m = -4 \) 35. No solutions 36. \( x^{8} y^{(b-2)} \)