%5E-3.jpeg "(10x^5y^3)^-3")
Simplifying Expressions with Negative Exponents: (10x^5y^3)^-3
In mathematics, understanding how to simplify expressions involving negative exponents is crucial. Let's break down the simplification of (10x^5y^3)^-3.
The Power of a Product Rule
The key to simplifying this expression lies in understanding the power of a product rule: (ab)^n = a^n * b^n.
This rule states that when raising a product to a power, we can distribute the power to each factor within the product.
Applying the Rule to our Expression
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Distribute the exponent: Applying the power of a product rule, we get: (10x^5y^3)^-3 = 10^-3 * (x^5)^-3 * (y^3)^-3
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Simplify further: Now we need to simplify each term using the power of a power rule: (a^m)^n = a^(m*n).
- 10^-3 = 1/10^3 = 1/1000
- (x^5)^-3 = x^(5*-3) = x^-15
- (y^3)^-3 = y^(3*-3) = y^-9
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Dealing with negative exponents: Remember that a negative exponent indicates the reciprocal of the base raised to the positive value of the exponent: a^-n = 1/a^n
- x^-15 = 1/x^15
- y^-9 = 1/y^9
The Final Simplified Expression
Putting everything together, the simplified expression of (10x^5y^3)^-3 is:
(10x^5y^3)^-3 = 1/1000 * 1/x^15 * 1/y^9 = 1/(1000x^15y^9)
Key Takeaways
- Power of a Product Rule: It allows us to distribute exponents to individual factors within a product.
- Power of a Power Rule: It helps simplify expressions with exponents raised to other exponents.
- Negative Exponents: They represent reciprocals of the base raised to the positive value of the exponent.
By applying these rules, we can confidently simplify complex expressions involving negative exponents.