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

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UpStudy AI · Accepted Answer

Simplify the expression by following steps:
- step0: Solution:
  $$2.3\left(81a^{16}b^{8}\right)^{\frac{3}{4}}$$
- step1: Rewrite the expression:
  $$2.3\times 27a^{12}b^{6}$$
- step2: Multiply the numbers:
  $$62.1a^{12}b^{6}$$
Calculate or simplify the expression $$ 2.2*(64*a^{6})^{2/3} $$.
Simplify the expression by following steps:
- step0: Solution:
  $$2.2\left(64a^{6}\right)^{\frac{2}{3}}$$
- step1: Rewrite the expression:
  $$2.2\times 16a^{4}$$
- step2: Multiply the numbers:
  $$35.2a^{4}$$
Calculate or simplify the expression $$ 21*(27*x^{27})^{1/3} $$.
Simplify the expression by following steps:
- step0: Solution:
  $$21\left(27x^{27}\right)^{\frac{1}{3}}$$
- step1: Rewrite the expression:
  $$21\times 3x^{9}$$
- step2: Multiply the numbers:
  $$63x^{9}$$
Calculate or simplify the expression $$ (a^{8}*b^{16}*c^{20})^{1/4} $$.
Simplify the expression by following steps:
- step0: Solution:
  $$\left(a^{8}b^{16}c^{20}\right)^{\frac{1}{4}}$$
- step1: Use the properties of exponents:
  $$\left(a^{8}\right)^{\frac{1}{4}}\left(b^{16}\right)^{\frac{1}{4}}\left(c^{20}\right)^{\frac{1}{4}}$$
- step2: Evaluate the power:
  $$a^{2}b^{4}c^{5}$$
Calculate or simplify the expression $$ (\sqrt[3]{x^{2}}*\sqrt[3]{x^{4}}) $$.
Simplify the expression by following steps:
- step0: Solution:
  $$\sqrt[3]{x^{2}}\times \sqrt[3]{x^{4}}$$
- step1: Use the properties of radicals:
  $$\sqrt[3]{x^{2}\times x^{4}}$$
- step2: Multiply the terms:
  $$\sqrt[3]{x^{6}}$$
- step3: Transform the expression:
  $$\sqrt[3]{\left(x^{2}\right)^{3}}$$
- step4: Simplify the root:
  $$x^{2}$$
Calculate or simplify the expression $$ (15*a^{15}+17*a^{15})^{1/5} $$.
Simplify the expression by following steps:
- step0: Solution:
  $$\left(15a^{15}+17a^{15}\right)^{\frac{1}{5}}$$
- step1: Add the terms:
  $$\left(32a^{15}\right)^{\frac{1}{5}}$$
- step2: Use the properties of exponents:
  $$32^{\frac{1}{5}}\left(a^{15}\right)^{\frac{1}{5}}$$
- step3: Evaluate the power:
  $$2a^{3}$$
Calculate or simplify the expression $$ (125*a^{12}*b^{15}*c^{21})^{1/3} $$.
Simplify the expression by following steps:
- step0: Solution:
  $$\left(125a^{12}b^{15}c^{21}\right)^{\frac{1}{3}}$$
- step1: Use the properties of exponents:
  $$125^{\frac{1}{3}}\left(a^{12}\right)^{\frac{1}{3}}\left(b^{15}\right)^{\frac{1}{3}}\left(c^{21}\right)^{\frac{1}{3}}$$
- step2: Evaluate the power:
  $$5a^{4}b^{5}c^{7}$$
Calculate or simplify the expression $$ (x^{5})^{1/3}*x^{1/3} $$.
Simplify the expression by following steps:
- step0: Solution:
  $$\left(x^{5}\right)^{\frac{1}{3}}x^{\frac{1}{3}}$$
- step1: Multiply the exponents:
  $$x^{5\times \frac{1}{3}}\times x^{\frac{1}{3}}$$
- step2: Multiply:
  $$x^{\frac{5}{3}}\times x^{\frac{1}{3}}$$
- step3: Simplify the expression:
  $$x^{\frac{5}{3}+\frac{1}{3}}$$
- step4: Add the numbers:
  $$x^{2}$$
Calculate or simplify the expression $$ (16/x^{-8})^{3/4} $$.
Simplify the expression by following steps:
- step0: Solution:
  $$\left(\frac{16}{x^{-8}}\right)^{\frac{3}{4}}$$
- step1: Rewrite the expression:
  $$\left(16x^{8}\right)^{\frac{3}{4}}$$
- step2: Use the properties of exponents:
  $$16^{\frac{3}{4}}\left(x^{8}\right)^{\frac{3}{4}}$$
- step3: Evaluate the power:
  $$8x^{6}$$
Calculate or simplify the expression $$ (\sqrt[3]{b^{2}}/b^{-3})^{2} $$.
Simplify the expression by following steps:
- step0: Solution:
  $$\left(\frac{\sqrt[3]{b^{2}}}{b^{-3}}\right)^{2}$$
- step1: Divide the terms:
  $$\left(b^{3}\sqrt[3]{b^{2}}\right)^{2}$$
- step2: Evaluate the power:
  $$\left(b^{3}\right)^{2}\left(\sqrt[3]{b^{2}}\right)^{2}$$
- step3: Evaluate the power:
  $$b^{6}\left(\sqrt[3]{b^{2}}\right)^{2}$$
- step4: Evaluate the power:
  $$b^{6}\times b\sqrt[3]{b}$$
- step5: Multiply the terms:
  $$b^{7}\sqrt[3]{b}$$
Calculate or simplify the expression $$ (\sqrt{x}/x^{-3/2})^{2} $$.
Simplify the expression by following steps:
- step0: Solution:
  $$\left(\frac{\sqrt{x}}{x^{\frac{-3}{2}}}\right)^{2}$$
- step1: Rewrite the fraction:
  $$\left(\frac{\sqrt{x}}{x^{-\frac{3}{2}}}\right)^{2}$$
- step2: Divide the terms:
  $$\left(x^{2}\right)^{2}$$
- step3: Transform the expression:
  $$x^{2\times 2}$$
- step4: Multiply the numbers:
  $$x^{4}$$
Calculate or simplify the expression $$ (\sqrt[4]{81*a^{-8}*b^{12}})/(1/(4*a^{2}*b^{-3}))^{-1} $$.
Simplify the expression by following steps:
- step0: Solution:
  $$\frac{\left(\sqrt[4]{81a^{-8}b^{12}}\right)}{\left(\frac{1}{\left(4a^{2}b^{-3}\right)}\right)^{-1}}$$
- step1: Remove the parentheses:
  $$\frac{\sqrt[4]{81a^{-8}b^{12}}}{\left(\frac{1}{4a^{2}b^{-3}}\right)^{-1}}$$
- step2: Rewrite the expression:
  $$\frac{\sqrt[4]{81a^{-8}b^{12}}}{\left(\frac{b^{3}}{4a^{2}}\right)^{-1}}$$
- step3: Simplify the root:
  $$\frac{\frac{3b^{3}}{a^{2}}}{\left(\frac{b^{3}}{4a^{2}}\right)^{-1}}$$
- step4: Rewrite the expression:
  $$\frac{\frac{3b^{3}}{a^{2}}}{\frac{4a^{2}}{b^{3}}}$$
- step5: Multiply by the reciprocal:
  $$\frac{3b^{3}}{a^{2}}\times \frac{b^{3}}{4a^{2}}$$
- step6: Multiply the terms:
  $$\frac{3b^{3}\times b^{3}}{a^{2}\times 4a^{2}}$$
- step7: Multiply the terms:
  $$\frac{3b^{6}}{a^{2}\times 4a^{2}}$$
- step8: Multiply the terms:
  $$\frac{3b^{6}}{4a^{4}}$$
Here are the simplified results for each expression:

1. **For** $$ 21\left(27 x^{27}\right)^{\frac{1}{3}} $$:
   $$
   63x^{9}
   $$

2. **For** $$ 2.2\left(64 a^{6}\right)^{\frac{2}{3}} $$:
   $$
   35.2a^{4}
   $$

3. **For** $$ 2.3 \quad\left(81 a^{16} b^{8}\right)^{\frac{3}{4}} $$:
   $$
   62.1a^{12}b^{6}
   $$

4. **For** $$ 2.4 \sqrt[4]{a^{8} b^{16} c^{20}} $$:
   $$
   a^{2}b^{4}c^{5}
   $$

5. **For** $$ 2.5 \sqrt[3]{125 a^{12} b^{15} c^{21}} $$:
   $$
   5a^{4}b^{5}c^{7}
   $$

6. **For** $$ 2.6 \sqrt[5]{15 a^{15}+17 a^{15}} $$:
   $$
   2a^{3}
   $$

7. **For** $$ 2.7 \quad\left(\frac{16}{x^{-8}}\right)^{\frac{3}{4}} $$:
   $$
   8x^{6}
   $$

8. **For** $$ 2.8\left(\frac{\sqrt{x}}{x^{-\frac{3}{2}}}\right)^{2} $$:
   $$
   x^{4}
   $$

9. **For** $$ 2.9 \sqrt[3]{x^{5}} \cdot x^{\frac{1}{3}} $$:
   $$
   x^{2}
   $$

10. **For** $$ 2.10\left(\frac{\sqrt[3]{b^{2}}}{b^{-3}}\right)^{2} $$:
    $$
    b^{7}\sqrt[3]{b}
    $$

11. **For** $$ 2.11 \sqrt[3]{x^{2}} \cdot \sqrt[3]{x^{4}} $$:
    $$
    x^{2}
    $$

12. **For** $$ 2.12 \frac{\sqrt[4]{81 a^{-8} b^{12}}}{\left(\frac{1}{4} a^{2} b^{-3}\right)^{-1}} $$:
    $$
    \frac{3b^{6}}{4a^{4}}
    $$

These results provide the simplified forms of the given expressions.
Here are the simplified results for each expression: 1. $$ 63x^{9} $$ 2. $$ 35.2a^{4} $$ 3. $$ 62.1a^{12}b^{6} $$ 4. $$ a^{2}b^{4}c^{5} $$ 5. $$ 5a^{4}b^{5}c^{7} $$ 6. $$ 2a^{3} $$ 7. $$ 8x^{6} $$ 8. $$ x^{4} $$ 9. $$ x^{2} $$ 10. $$ b^{7}\sqrt[3]{b} $$ 11. $$ x^{2} $$ 12. $$ \frac{3b^{6}}{4a^{4}} $$

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