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Simplify and give reasons (5/6)³ ×(6/5)-8​

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Final answer:

To simplify the expression (5/6)³ × (6/5)-8, we first raise each fraction to the given powers. Then, we multiply the resulting fractions together and simplify. The final simplified expression is 1/5³².

Step-by-step explanation:

To simplify the expression (5/6)³ × (6/5)-8, we need to work with exponents and multiply the fractions. Here's a step-by-step explanation:

  1. Start by simplifying (5/6)³. This means raising both the numerator and denominator to the power of 3. (5/6)³ becomes (5³)/(6³) which is equal to 125/216.
  2. Next, simplify (6/5)-8. This means raising both the numerator and denominator to the power of -8. (6/5)-8 becomes (6⁻⁸)/(5⁻⁸) which is equal to (1/(6⁸))/(1/(5⁸)).
  3. Now, multiply the two fractions together: (125/216) * [(1/(6⁸))/(1/(5⁸))]. To multiply fractions, multiply the numerators together and multiply the denominators together. This gives us [125/(6⁸)] * [(5⁸)/216].
  4. Finally, simplify the expression [125/(6⁸)] * [(5⁸)/216]. To simplify, cancel out common factors. In this case, we can cancel out the '6⁸' in the numerator with the '216' in the denominator. This leaves us with 125/5⁸, which simplifies further to 1/5³².

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