Simplify each exponential expression in Exercises 23–64. $(- 3 x^{2} y^{5})^{2}$

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Identify the expression to simplify: $(-3x^2 y^5)^2$. This means the entire quantity inside the parentheses is raised to the power of 2.

Apply the power of a product rule, which states that $(abc)^n = a^n b^n c^n$. So, rewrite the expression as $(-3)^2 (x^2)^2 (y^5)^2$.

Simplify each part separately: $(-3)^2$ means multiply -3 by itself, $(x^2)^2$ means raise $x^2$ to the power 2, and $(y^5)^2$ means raise $y^5$ to the power 2.

Use the power of a power rule for the variables: $(x^2)^2 = x^{2 \times 2} = x^4$ and $(y^5)^2 = y^{5 \times 2} = y^{10}$.

Combine all simplified parts to write the final expression as $(-3)^2 x^4 y^{10}$.

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Exponentiation of a Product

When raising a product to a power, each factor inside the parentheses is raised to that power separately. For example, (ab)^n = a^n * b^n. This rule helps simplify expressions like (−3x^2 y^5)^2 by applying the exponent to −3, x^2, and y^5 individually.

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Power of a Power Rule

The power of a power rule states that (a^m)^n = a^(m*n). This means when an exponent is raised to another exponent, you multiply the exponents. In the expression (x^2)^2, you multiply 2 by 2 to get x^4.

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Handling Negative Bases with Exponents

When a negative number is raised to an even power, the result is positive because multiplying an even number of negative factors yields a positive product. For example, (−3)^2 = 9. This is important for correctly simplifying expressions like (−3x^2 y^5)^2.

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