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. \( x^{3} y^{\text {b }} \cdot x^{5}-y^{2} \)

**31.** We have \[ 5^2 \cdot 5^m = 5^{2+m}, \] and since this equals \(5^{11}\) we set \[ 2+m=11. \] Thus, \[ m=9. \] **32.** The equation is \[ m^\oplus \cdot m^{-4} = m^{-9}. \] Recall that when multiplying powers with the same base we add the exponents. Denote the unknown exponent by \(n\) so that \[ m^n \cdot m^{-4}=m^{n-4}. \] Setting the exponent equal to \(-9\) gives \[ n-4=-9. \] Thus, \[ n=-5. \] In this context, the symbol \(\oplus\) represents the exponent \(-5\). **33.** The product is \[ 2^{(m-2)} \cdot 2^1=2^{m-2+1}=2^{m-1}. \] **34.** The equation is \[ a^m \cdot a^4 = a^{m+4}=1. \] Assuming \(a\) is nonzero and not equal to 1, the only way a power of \(a\) equals 1 is when the exponent is zero. Therefore, we require \[ m+4=0. \] So, \[ m=-4. \] **35.** The equation is \[ a^l \cdot a^{-2}=a^{l-2}, \] and it is given that \[ a^{l-2}=a^l. \] For nonzero \(a\) (with \(a\neq1\)) we equate exponents: \[ l-2=l. \] Subtracting \(l\) from both sides yields \[ -2=0, \] which is impossible. Hence, for the equality to hold the only possibility is if the base \(a\) is such that \[ a^k=1, \] for every exponent \(k\). For example, if \(a=1\) then the equation holds for any \(l\). Otherwise, there is no solution. **36.** The expression is \[ x^3 y^b \cdot x^5 - y^2. \] Given the context of the previous problems, it is likely that the intended multiplication involves both the \(x\) and \(y\) terms. Interpreting the expression as \[ x^3 y^b \cdot x^5 y^{-2}, \] we use the properties of exponents to combine like bases: \[ x^{3+5} y^{b+(-2)} = x^8 y^{b-2}. \] Thus, the complete expression is \[ x^8 y^{b-2}. \]