Simplify the following exponential expression.

Question

StartFraction 243 y Superscript 5 Baseline z Superscript 8 Over negative 27 y Superscript 9 Baseline z Superscript negative 15 EndFraction
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Part 1
StartFraction 243 y Superscript 5 Baseline z Superscript 8 Over negative 27 y Superscript 9 Baseline z Superscript negative 15 EndFraction
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enter your response here   ​(Simplify your​ answer.)

Answer 1

To simplify the expression

\[ \frac{243 y^5 z^8}{-27 y^9 z^{-15}}, \]

we will proceed with the simplification step by step.

1. Simplify the numerical coefficients:

   The coefficient in the numerator is \(243\) and in the denominator is \(-27\).

   \[ \frac{243}{-27} = -9. \]

2. Simplify the \(y\) terms:

   We have \(y^5\) in the numerator and \(y^9\) in the denominator.

   Using the property of exponents \(\frac{a^m}{a^n} = a^{m-n}\), we get:

   \[ \frac{y^5}{y^9} = y^{5-9} = y^{-4}. \]

3. Simplify the \(z\) terms:

   We have \(z^8\) in the numerator and \(z^{-15}\) in the denominator.

   Applying the same property, we have:

   \[ \frac{z^8}{z^{-15}} = z^{8 - (-15)} = z^{8 + 15} = z^{23}. \]

4. Combine all parts:

   Now we can combine the results of our simplifications:

   \[ -9 y^{-4} z^{23}. \]

5. Rewrite \(y^{-4}\) (if needed):

   If we want to express \(y^{-4}\) in a different way, we can write it as:

   \[ \frac{-9 z^{23}}{y^4}. \]

Thus, the simplified expression, in either form, is:

\[ -9 y^{-4} z^{23} \quad \text{or} \quad \frac{-9 z^{23}}{y^4}. \]