X ^ 11 * x ^ 12 x ^ 9 * y ^ 9 6) (x ^ 21) / (x ^ 14) (x/y) ^ 6 B) (x ^ 8) ^ 4 e) (a/3) ^ 7 * (3/a) ^ 7

Let's simplify each of the expressions one by one. 1. **\(x^{11} \cdot x^{12}\)**: Using the exponent rule that states \(a^m \cdot a^n = a^{m+n}\): \[ x^{11} \cdot x^{12} = x^{11+12} = x^{23} \] 2. **\(x^{9} \cdot y^{9}\)**: This expression is already in its simplest form, but if you wanted to factor it, it could be written as: \[ x^9 \cdot y^9 = (xy)^9 \] 3. **\(\frac{x^{21}}{x^{14}}\)**: Using the exponent rule that states \(\frac{a^m}{a^n} = a^{m-n}\): \[ \frac{x^{21}}{x^{14}} = x^{21-14} = x^{7} \] 4. **\(\left(\frac{x}{y}\right)^{6}\)**: This is already in its simplest form, but you can express it as: \[ \left(\frac{x}{y}\right)^{6} = \frac{x^6}{y^6} \] 5. **\((x^8)^4\)**: Using the exponent rule that states \((a^m)^n = a^{m \cdot n}\): \[ (x^8)^4 = x^{8 \cdot 4} = x^{32} \] 6. **\(\left(\frac{a}{3}\right)^{7} \cdot \left(\frac{3}{a}\right)^{7}\)**: Using the property that \((ab)^m = a^m b^m\) and the fact that the bases are the same: \[ \left(\frac{a}{3}\right)^{7} \cdot \left(\frac{3}{a}\right)^{7} = \left(\frac{a}{3} \cdot \frac{3}{a}\right)^{7} \] Simplifying inside the parentheses: \[ \frac{a \cdot 3}{3 \cdot a} = 1 \] Thus: \[ 1^{7} = 1 \] In summary, the simplified forms are: 1. \(x^{23}\) 2. \(x^{9} y^{9} = (xy)^{9}\) 3. \(x^{7}\) 4. \(\frac{x^{6}}{y^{6}}\) 5. \(x^{32}\) 6. \(1\)