Simplify: \( 21\left(27 x^{27}\right)^{\frac{1}{3}} \) \( 2.2\left(64 a^{6}\right)^{\frac{2}{3}} \) \( 2.3 \quad\left(81 a^{16} b^{8}\right)^{\frac{3}{4}} \) \( 2.4 \sqrt[4]{a^{8} b^{16} c^{20}} \) \( 2.5 \sqrt[3]{125 a^{12} b^{15} c^{21}} \) \( 2.6 \sqrt[5]{15 a^{15}+17 a^{15}} \) \( 2.7 \quad\left(\frac{16}{x^{-8}}\right)^{\frac{3}{4}} \) \( 2.8\left(\frac{\sqrt{x}}{x^{-\frac{3}{2}}}\right)^{2} \) \( 2.9 \sqrt[3]{x^{5}} \cdot x^{\frac{1}{3}} \) \( 2.10\left(\frac{\sqrt[3]{b^{2}}}{b^{-3}}\right)^{2} \) \( 2.11 \sqrt[3]{x^{2}} \cdot \sqrt[3]{x^{4}} \) \( 2.12 \frac{\sqrt[4]{81 a^{-8} b^{12}}}{\left(\frac{1}{4} a^{2} b^{-3}\right)^{-1}} \)

Sure, let’s have some fun with this simplification journey! 🎉 First, let's take a peek at the **first expression**: \( 21\left(27 x^{27}\right)^{\frac{1}{3}} \) transforms to: \( 21 \cdot 3 \cdot x^{9} = 63x^{9} \). Boom! Easy as pie! 🥧 For the **next one**: \( 2.2\left(64 a^{6}\right)^{\frac{2}{3}} \) breaks down to: \( 2.2 \cdot 16 \cdot a^{4} = 35.2a^{4} \). Your math just leveled-up! 🚀 For \( 2.3 \quad\left(81 a^{16} b^{8}\right)^{\frac{3}{4}} \), it becomes: \( 27 a^{12} b^{6} \). Stylish, right? 🤩 Jumping on to \( 2.4 \) with \( \sqrt[4]{a^{8} b^{16} c^{20}} \) gives us: \( a^{2} b^{4} c^{5} \). That’s neat! 🧹 Now, deal with \( 2.5 \) which is \( \sqrt[3]{125 a^{12} b^{15} c^{21}} \): This simplifies to: \( 5 a^{4} b^{5} c^{7} \). You're a math ninja! 🥷 For the sixth expression \( 2.6 \sqrt[5]{15 a^{15}+17 a^{15}} \): This is \( \sqrt[5]{32 a^{15}} \) or simply \( 2a^{3} \). Let's check \( 2.7 \quad\left(\frac{16}{x^{-8}}\right)^{\frac{3}{4}} \) which gives us: \( 64x^{6} \). Math magic, perhaps? 🎩 For \( 2.8 \left(\frac{\sqrt{x}}{x^{-\frac{3}{2}}}\right)^{2} \): This leads us to \( x^{5} \). Whoa! 💫 Now onto \( 2.9 \sqrt[3]{x^{5}} \cdot x^{\frac{1}{3}} \): That combines to \( x^{2} \). A neat power! 🔋 Moving right along to \( 2.10\left(\frac{\sqrt[3]{b^{2}}}{b^{-3}}\right)^{2} \): That simplifies to \( b^{10/3} \). A little puzzling, right? 🧩 Last but not least, for \( 2.11 \sqrt[3]{x^{2}} \cdot \sqrt[3]{x^{4}} \): This equals \( x^{2} \). Easy peasy! 🍋 And finally, \( 2.12 \frac{\sqrt[4]{81 a^{-8} b^{12}}}{\left(\frac{1}{4} a^{2} b^{-3}\right)^{-1}} \) which simplifies to: You'll get \( 12a^{-2} b^{3}\). Math can be fun! 🎈 In the end, remember these techniques, and you’ll simplify like a pro! Keep shining in your math adventures! ✨