Question

Simplify the following exponential expression.
StartFraction 243 y Superscript 5 Baseline z Superscript 8 Over negative 27 y Superscript 9 Baseline z Superscript negative 15 EndFraction
Question content area bottom
Part 1
StartFraction 243 y Superscript 5 Baseline z Superscript 8 Over negative 27 y Superscript 9 Baseline z Superscript negative 15 EndFraction
equals
  
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:

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