Sure! Let's go through a step-by-step solution for simplifying the given expression and finding the sum of the constants [tex]\(c\)[/tex] and [tex]\(d\)[/tex].
We start with the expression:
[tex]\[
\frac{32 x^6}{4 x^3}
\][/tex]
1. Combine the terms in the fraction:
We can simplify this expression by combining the constants and the terms involving [tex]\(x\)[/tex]:
[tex]\[
\frac{32 x^6}{4 x^3} = \frac{32}{4} \cdot \frac{x^6}{x^3}
\][/tex]
2. Simplify the constants:
Simplify the numerical portion [tex]\(\frac{32}{4}\)[/tex]:
[tex]\[
\frac{32}{4} = 8
\][/tex]
3. Simplify the exponents:
Simplify the terms involving [tex]\(x\)[/tex]:
[tex]\[
\frac{x^6}{x^3} = x^{6-3} = x^3
\][/tex]
4. Combine the results:
Now, we combine the simplified constant term with the simplified term involving [tex]\(x\)[/tex]:
[tex]\[
8 \cdot x^3
\][/tex]
Thus, the expression [tex]\(\frac{32 x^6}{4 x^3}\)[/tex] simplifies to [tex]\(8x^3\)[/tex].
5. Identify constants [tex]\(c\)[/tex] and [tex]\(d\)[/tex]:
From the simplified expression [tex]\(8x^3\)[/tex], it is clear that:
[tex]\[
c = 8 \quad \text{(the coefficient of } x^3\text{)}
\][/tex]
[tex]\[
d = 3 \quad \text{(the exponent of } x\text{)}
\][/tex]
6. Calculate [tex]\(c + d\)[/tex]:
Finally, we add the constants [tex]\(c\)[/tex] and [tex]\(d\)[/tex]:
[tex]\[
c + d = 8 + 3 = 11
\][/tex]
So, the value of [tex]\(c + d\)[/tex] is [tex]\(11\)[/tex].
