Wood Weston
06/13/2024 · Elementary School
Simplify: \( \left(7^{\frac{5}{7}}\right)\left(7^{\frac{7}{5}}\right) \) Give an exact answer in both rational exponent form and nth root notation. Rational Exponents: nth root notation: Decimal Approximation: Round your answer to 3 decimal places
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Step-by-step Solution
Calculate or simplify the expression \( (7^(5/7))*(7^(7/5)) \).
Calculate the value by following steps:
- step0: Calculate:
\(7^{\frac{5}{7}}\times 7^{\frac{7}{5}}\)
- step1: Multiply:
\(7^{\frac{5}{7}+\frac{7}{5}}\)
- step2: Add the terms:
\(7^{\frac{74}{35}}\)
- step3: Transform the expression:
\(\sqrt[35]{7^{74}}\)
- step4: Rewrite the expression:
\(\sqrt[35]{7^{70}\times 7^{4}}\)
- step5: Rewrite the expression:
\(\sqrt[35]{7^{70}}\times \sqrt[35]{7^{4}}\)
- step6: Rewrite the expression:
\(7^{2}\sqrt[35]{7^{4}}\)
- step7: Calculate:
\(49\sqrt[35]{7^{4}}\)
- step8: Evaluate:
\(49\sqrt[35]{2401}\)
The simplified form of \( \left(7^{\frac{5}{7}}\right)\left(7^{\frac{7}{5}}\right) \) is \( 49\sqrt[35]{2401} \).
In nth root notation, this is \( 49\sqrt[35]{2401} \).
The decimal approximation of this expression is approximately 61.204.
Quick Answer
Rational Exponents: \( 49 \)
nth root notation: \( \sqrt[35]{2401} \)
Decimal Approximation: 61.204
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