Certainly! Let's solve the problem step-by-step to show that [tex]\(\frac{x}{z} + \frac{y}{z} = 4b + 2a\)[/tex] given the expressions [tex]\(x = 8a^4b^5\)[/tex], [tex]\(y = 4a^5b^4\)[/tex], and [tex]\(z = 2a^4b^4\)[/tex].
1. Calculate [tex]\(\frac{x}{z}\)[/tex]:
Given [tex]\(x = 8a^4b^5\)[/tex] and [tex]\(z = 2a^4b^4\)[/tex]:
[tex]\[
\frac{x}{z} = \frac{8a^4b^5}{2a^4b^4}
\][/tex]
Simplify the fraction by canceling out the common terms in the numerator and denominator:
[tex]\[
\frac{x}{z} = \frac{8a^4b^5}{2a^4b^4} = \frac{8}{2} \cdot \frac{a^4}{a^4} \cdot \frac{b^5}{b^4} = 4 \cdot 1 \cdot b = 4b
\][/tex]
2. Calculate [tex]\(\frac{y}{z}\)[/tex]:
Given [tex]\(y = 4a^5b^4\)[/tex] and [tex]\(z = 2a^4b^4\)[/tex]:
[tex]\[
\frac{y}{z} = \frac{4a^5b^4}{2a^4b^4}
\][/tex]
Simplify the fraction by canceling out the common terms in the numerator and denominator:
[tex]\[
\frac{y}{z} = \frac{4a^5b^4}{2a^4b^4} = \frac{4}{2} \cdot \frac{a^5}{a^4} \cdot \frac{b^4}{b^4} = 2 \cdot a \cdot 1 = 2a
\][/tex]
3. Sum the simplified fractions:
Now, add the two simplified fractions together:
[tex]\[
\frac{x}{z} + \frac{y}{z} = 4b + 2a
\][/tex]
Therefore, we have shown that [tex]\(\frac{x}{z} + \frac{y}{z} = 4b + 2a\)[/tex].
