To simplify the expression [tex]\(\frac{(x y)^7}{y^6} \cdot \frac{5}{x^8}\)[/tex], follow these steps:
### Step 1: Simplify the Fraction [tex]\(\frac{(x y)^7}{y^6}\)[/tex]
First, focus on the left part of the expression [tex]\(\frac{(x y)^7}{y^6}\)[/tex]:
[tex]\[ \frac{(xy)^7}{y^6} = \frac{x^7 y^7}{y^6} \][/tex]
Since we have [tex]\(y^7\)[/tex] in the numerator and [tex]\(y^6\)[/tex] in the denominator, we can simplify by subtracting the exponent in the denominator from the exponent in the numerator (using the rule [tex]\(\frac{a^m}{a^n} = a^{m-n}\)[/tex]):
[tex]\[ \frac{x^7 y^7}{y^6} = x^7 y^{7-6} = x^7 y^1 = x^7 y \][/tex]
### Step 2: Multiply by the Second Fraction [tex]\(\frac{5}{x^8}\)[/tex]
Next, we multiply the simplified result from Step 1 by the second fraction [tex]\(\frac{5}{x^8}\)[/tex]:
[tex]\[ x^7 y \cdot \frac{5}{x^8} = \frac{x^7 y \cdot 5}{x^8} = \frac{5x^7 y}{x^8} \][/tex]
### Step 3: Simplify the Resultant Expression
Now, simplify the fraction [tex]\(\frac{5x^7 y}{x^8}\)[/tex]:
[tex]\[ \frac{5x^7 y}{x^8} = 5 y \cdot \frac{x^7}{x^8} \][/tex]
Using the same exponent rule as before, [tex]\(\frac{x^7}{x^8} = x^{7-8} = x^{-1}\)[/tex]:
[tex]\[ 5 y \cdot x^{-1} = \frac{5y}{x} \][/tex]
### Final Result
The simplified expression is:
[tex]\[ \frac{5y}{x} \][/tex]
So, the final simplified form of the given expression [tex]\(\frac{(x y)^7}{y^6} \cdot \frac{5}{x^8}\)[/tex] is:
[tex]\[ \boxed{\frac{5y}{x}} \][/tex]
