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Given the algebraic expression the quantity 125 times x to the seven-thirds power end quantity to the negative one-third power comma, create an equivalent expression.

a 5 times x to the seven-ninths power
b 5x2
c 1 divided by the quantity 5 times x to the seven-ninths power end quantity
d 1 divided by the quantity 5 times x squared end quantity

1 answer

To simplify the expression \( (125x^{7/3})^{-1/3} \), we can follow these steps:

  1. Apply the negative exponent rule: \( a^{-n} = \frac{1}{a^n} \). So: \[ (125x^{7/3})^{-1/3} = \frac{1}{(125x^{7/3})^{1/3}} \]

  2. Now simplify the expression inside the denominator: \[ (125x^{7/3})^{1/3} = 125^{1/3} \cdot (x^{7/3})^{1/3} \]

  3. Calculate \( 125^{1/3} \): \[ 125^{1/3} = 5 \]

  4. Simplify \( (x^{7/3})^{1/3} \) using the property \( (a^{m})^{n} = a^{mn} \): \[ (x^{7/3})^{1/3} = x^{(7/3) \cdot (1/3)} = x^{7/9} \]

  5. So we have: \[ (125x^{7/3})^{1/3} = 5x^{7/9} \]

  6. Putting it all together, we find: \[ (125x^{7/3})^{-1/3} = \frac{1}{5x^{7/9}} \]

Thus, the expression simplifies to \( \frac{1}{5x^{7/9}} \).

The answer that matches our result is:

c \( \frac{1}{5x^{7/9}} \)