Question

Simplify the exponential expression. Assume that the variable represents a nonzero real number.
StartFraction 24 z cubed Over 12 z Superscript 10 EndFraction
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Part 1
StartFraction 24 z cubed Over 12 z Superscript 10 EndFraction
equals
  
enter your response here ​(Type exponential notation with positive​ exponents.)

Answer 1

To simplify the expression \(\frac{24 z^3}{12 z^{10}}\), we can start by simplifying the coefficients and then the exponential part separately.

1. **Simplify the coefficients (24 and 12)**:
\[
\frac{24}{12} = 2
\]

2. **Simplify the \(z\) terms**:
Here we can use the rule of exponents which states that \(\frac{a^m}{a^n} = a^{m-n}\).
\[
\frac{z^3}{z^{10}} = z^{3-10} = z^{-7}
\]

Now combine the simplified parts:
\[
\frac{24 z^3}{12 z^{10}} = 2 \cdot z^{-7} = 2 z^{-7}
\]

Since the problem asks for positive exponents, we can rewrite \(z^{-7}\) as \(\frac{1}{z^7}\):
\[
2 z^{-7} = \frac{2}{z^7}
\]

Thus, the final simplified expression is:
\[
\frac{2}{z^7}
\]

Or, in exponential notation:
\[
2 z^{-7}
\]

So either \(\frac{2}{z^7}\) or \(2 z^{-7}\) are acceptable simplified forms.