%5E3%2F(-a%5E4b%5E2)%5E5.jpeg "(-2ab^7)^3/(-a^4b^2)^5")
Simplifying the Expression: (-2ab^7)^3 / (-a^4b^2)^5
This problem involves simplifying an expression with exponents. To do so, we'll use the following rules:
Rules of Exponents:
- Power of a Product: (ab)^n = a^n * b^n
- Power of a Quotient: (a/b)^n = a^n / b^n
- Power of a Power: (a^m)^n = a^(m*n)
Step 1: Expand the powers
Apply the "Power of a Product" rule to both the numerator and denominator:
- Numerator: (-2ab^7)^3 = (-2)^3 * a^3 * (b^7)^3
- Denominator: (-a^4b^2)^5 = (-1)^5 * (a^4)^5 * (b^2)^5
Step 2: Simplify the exponents
Apply the "Power of a Power" rule to simplify the exponents:
- Numerator: (-2)^3 * a^3 * (b^7)^3 = -8 * a^3 * b^(7*3) = -8a^3b^21
- Denominator: (-1)^5 * (a^4)^5 * (b^2)^5 = -1 * a^(45) * b^(25) = -a^20b^10
Step 3: Combine the results
Now we have: (-8a^3b^21) / (-a^20b^10)
Step 4: Simplify using Quotient Rule
Apply the "Power of a Quotient" rule:
- (-8a^3b^21) / (-a^20b^10) = (-8 / -1) * (a^3 / a^20) * (b^21 / b^10)
Step 5: Final Simplification
Simplify each term:
- 8 * a^(3-20) * b^(21-10) = 8a^-17b^11
Therefore, the simplified expression is 8a^-17b^11.