Answer :

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].