To simplify the expression:
[tex]\[
\frac{24 u^3}{64 u}
\][/tex]
we will follow these steps:
1. Factor out the numerators and denominators: First, we recognize that both the numerator and the denominator can be expressed in terms of their factors.
The numerator [tex]\(24 u^3\)[/tex] can be factored as:
[tex]\[
24 u^3 = 24 \cdot u^3
\][/tex]
The denominator [tex]\(64 u\)[/tex] can be factored as:
[tex]\[
64 u = 64 \cdot u
\][/tex]
2. Cancel the common factors: We notice that [tex]\(u\)[/tex] appears in both the numerator and the denominator. We can cancel out one [tex]\(u\)[/tex] from each:
[tex]\[
\frac{24 \cdot u^3}{64 \cdot u} = \frac{24 \cdot u^2 \cdot u}{64 \cdot u} = \frac{24 \cdot u^2}{64}
\][/tex]
3. Simplify the coefficient fraction: Now we simplify the fraction [tex]\(\frac{24}{64}\)[/tex]. We can find the greatest common divisor (GCD) of 24 and 64, which is 8. Both 24 and 64 are divisible by 8:
[tex]\[
\frac{24}{64} = \frac{24 \div 8}{64 \div 8} = \frac{3}{8}
\][/tex]
4. Write the final simplified form:
[tex]\[
\frac{3 \cdot u^2}{8}
\][/tex]
Thus, the simplified expression is:
[tex]\[
\frac{3 u^2}{8}
\][/tex]
