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Simplify [4]81x-8z4 and [3]256a6b12c9

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3 parts
i

3zx2\frac{3z}{x^2}

ii

4a2b4c3434a^2b^4c^3\sqrt[3]{4}

vii
6

Step-by-step solution

11 steps

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1. Simplify

Part i· 3 steps

81x−8z44\sqrt[4]{81x^{-8}z^4}

  1. Apply the property abcn=anbncn\sqrt[n]{abc} = \sqrt[n]{a}\sqrt[n]{b}\sqrt[n]{c}.

    814×x−84×z44\sqrt[4]{81} \times \sqrt[4]{x^{-8}} \times \sqrt[4]{z^4}

  2. Evaluate each term.

    814=3(since 34=81)\sqrt[4]{81} = 3 \quad (\text{since } 3^4 = 81) x−84=x−8/4=x−2\sqrt[4]{x^{-8}} = x^{-8/4} = x^{-2} z44=z\sqrt[4]{z^4} = z

  3. Combine the simplified terms.

    3×x−2×z=3zx23 \times x^{-2} \times z = \frac{3z}{x^2}

    The simplified expression is 3zx2\boxed{\frac{3z}{x^2}}.

Part ii

4a2b4c3434a^2b^4c^3\sqrt[3]{4}

256a6b12c93\sqrt[3]{256a^6b^{12}c^9}

  1. Factorize the constant term 256 to find its perfect cube factors.

    256=64×4=43×4256 = 64 \times 4 = 4^3 \times 4

  2. Apply the property abcn=anbncn\sqrt[n]{abc} = \sqrt[n]{a}\sqrt[n]{b}\sqrt[n]{c} and xmn=xm/n\sqrt[n]{x^m} = x^{m/n}.

    64×4×a6×b12×c93\sqrt[3]{64 \times 4 \times a^6 \times b^{12} \times c^9} 643×43×a63×b123×c93\sqrt[3]{64} \times \sqrt[3]{4} \times \sqrt[3]{a^6} \times \sqrt[3]{b^{12}} \times \sqrt[3]{c^9}

  3. Evaluate each term.

    643=4(since 43=64)\sqrt[3]{64} = 4 \quad (\text{since } 4^3 = 64) 43\sqrt[3]{4} a63=a6/3=a2\sqrt[3]{a^6} = a^{6/3} = a^2 b123=b12/3=b4\sqrt[3]{b^{12}} = b^{12/3} = b^4 c93=c9/3=c3\sqrt[3]{c^9} = c^{9/3} = c^3

  4. Combine the simplified terms.

    4×43×a2×b4×c3=4a2b4c3434 \times \sqrt[3]{4} \times a^2 \times b^4 \times c^3 = 4a^2b^4c^3\sqrt[3]{4}

    The simplified expression is 4a2b4c343\boxed{4a^2b^4c^3\sqrt[3]{4}}.

Part vii6

(216)23×(125)2(0.04)−3\sqrt{\frac{(216)^{\frac{2}{3}} \times (125)^2}{(0.04)^{-3}}}

  1. Simplify each term in the expression.

    (216)23=(63)23=63×23=62=36(216)^{\frac{2}{3}} = (6^3)^{\frac{2}{3}} = 6^{3 \times \frac{2}{3}} = 6^2 = 36 (125)2=(53)2=56(125)^2 = (5^3)^2 = 5^6 (0.04)−3=(4100)−3=(125)−3=(25)3=(52)3=56(0.04)^{-3} = \left(\frac{4}{100}\right)^{-3} = \left(\frac{1}{25}\right)^{-3} = (25)^3 = (5^2)^3 = 5^6

  2. Substitute the simplified terms back into the square root.

    36×5656\sqrt{\frac{36 \times 5^6}{5^6}}

  3. Cancel out common terms.

    36\sqrt{36}

  4. Evaluate the square root.

    36=6\sqrt{36} = 6

    The simplified expression is 6\boxed{6}.

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