Questions: (x^4 y^3 z^5)^6 / (x^5 y^3 z)^5

Transcript text: \[ \frac{\left(x^{4} y^{3} z^{5}\right)^{6}}{\left(x^{5} y^{3} z\right)^{5}} \]

Solution

Solution Steps

Solution Approach

Simplify the given expression by applying the power rule \((a^m)^n = a^{m \cdot n}\) to both the numerator and the denominator.
Use the quotient rule \(a^m / a^n = a^{m-n}\) to simplify the resulting expression.

Step 1: Simplifying the Numerator

The numerator of the expression is given by: \[ \left(x^{4} y^{3} z^{5}\right)^{6} = x^{24} y^{18} z^{30} \]

Step 2: Simplifying the Denominator

The denominator of the expression is: \[ \left(x^{5} y^{3} z\right)^{5} = x^{25} y^{15} z^{5} \]

Step 3: Forming the Expression

Now, we form the expression by dividing the simplified numerator by the simplified denominator: \[ \frac{x^{24} y^{18} z^{30}}{x^{25} y^{15} z^{5}} \]

Step 4: Applying the Quotient Rule

Using the quotient rule \(a^m / a^n = a^{m-n}\), we simplify each variable: \[ = x^{24-25} y^{18-15} z^{30-5} = x^{-1} y^{3} z^{25} \]

Step 5: Final Simplified Expression

Thus, the final simplified expression is: \[ \frac{y^{3} z^{25}}{x} \]