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Do not round any intermediate computations. \[ \begin{array}{c}\left(\frac{2}{7}\right)^{1.4}=\square \\ 0.3^{-0.35}=\square\end{array} \]

Ask by Herrera Salazar. in the United States
Feb 19,2025

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Answer

\( \left(\frac{2}{7}\right)^{1.4} \approx 0.173103 \) and \( 0.3^{-0.35} \approx 1.524079 \).

Solution

Calculate or simplify the expression \( (2/7)^{1.4} \). Calculate the value by following steps: - step0: Calculate: \(\left(\frac{2}{7}\right)^{1.4}\) - step1: Convert the expressions: \(\left(\frac{2}{7}\right)^{\frac{7}{5}}\) - step2: Use the properties of exponents: \(\frac{2^{\frac{7}{5}}}{7^{\frac{7}{5}}}\) - step3: Rewrite the expression: \(\frac{2^{\frac{7}{5}}}{7\sqrt[5]{7^{2}}}\) - step4: Rewrite the expression: \(\frac{2\sqrt[5]{2^{2}}}{7\sqrt[5]{7^{2}}}\) - step5: Multiply by the Conjugate: \(\frac{2\sqrt[5]{2^{2}}\times \sqrt[5]{7^{3}}}{7\sqrt[5]{7^{2}}\times \sqrt[5]{7^{3}}}\) - step6: Simplify: \(\frac{2\sqrt[5]{2^{2}}\times \sqrt[5]{343}}{7\sqrt[5]{7^{2}}\times \sqrt[5]{7^{3}}}\) - step7: Multiply the numbers: \(\frac{2\sqrt[5]{1372}}{7\sqrt[5]{7^{2}}\times \sqrt[5]{7^{3}}}\) - step8: Multiply the numbers: \(\frac{2\sqrt[5]{1372}}{49}\) Calculate or simplify the expression \( 0.3^{-0.35} \). Calculate the value by following steps: - step0: Calculate: \(0.3^{-0.35}\) - step1: Convert the expressions: \(\left(\frac{3}{10}\right)^{-\frac{7}{20}}\) - step2: Transform the expression: \(\left(\frac{10}{3}\right)^{\frac{7}{20}}\) - step3: Use the properties of exponents: \(\frac{10^{\frac{7}{20}}}{3^{\frac{7}{20}}}\) - step4: Rewrite the expression: \(\frac{10^{\frac{7}{20}}}{\sqrt[20]{3^{7}}}\) - step5: Rewrite the expression: \(\frac{\sqrt[20]{10^{7}}}{\sqrt[20]{3^{7}}}\) - step6: Multiply by the Conjugate: \(\frac{\sqrt[20]{10^{7}}\times \sqrt[20]{3^{13}}}{\sqrt[20]{3^{7}}\times \sqrt[20]{3^{13}}}\) - step7: Multiply the numbers: \(\frac{\sqrt[20]{10^{7}\times 3^{13}}}{\sqrt[20]{3^{7}}\times \sqrt[20]{3^{13}}}\) - step8: Multiply the numbers: \(\frac{\sqrt[20]{10^{7}\times 3^{13}}}{3}\) The results are: 1. \( \left(\frac{2}{7}\right)^{1.4} \approx 0.173103 \) 2. \( 0.3^{-0.35} \approx 1.524079 \)

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First, let's calculate \(\left(\frac{2}{7}\right)^{1.4}\). Using the property of exponents, we find: \[ \left(\frac{2}{7}\right)^{1.4} = \left(\frac{2}{7}\right)^{\frac{14}{10}} = \left(\frac{2}{7}\right)^{14} \cdot \left(\frac{2}{7}\right)^{-10} \] Calculating the power with a calculator gives approximately: \[ \left(\frac{2}{7}\right)^{1.4} \approx 0.2198 \] Now, let's calculate \(0.3^{-0.35}\). Using the property of negative exponents: \[ 0.3^{-0.35} = \frac{1}{0.3^{0.35}} \] We calculate \(0.3^{0.35}\) which gives us approximately: \[ 0.3^{0.35} \approx 0.7598 \] Thus, we have: \[ 0.3^{-0.35} \approx \frac{1}{0.7598} \approx 1.3169 \] Final results: \(\left(\frac{2}{7}\right)^{1.4} \approx 0.2198\) and \(0.3^{-0.35} \approx 1.3169\).

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