Simplify each exponential expression: (−5x3y2)(−2x−11y−$$2)(-5x^3y^2$)(-2x^{-11}$y^{-2})$$

Ch. P - Fundamental Concepts of Algebra

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Blitzer 8th Edition

Ch. P - Fundamental Concepts of Algebra

문제 29

1장, 문제 29

Simplify each exponential expression: $(- 5 x^{3} y^{2}) (- 2 x^{- 11} y^{- 2})$

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Identify the expression to simplify: $(-5x^{3}y^{2})(-2x^{-11}y^{-2})$.

Multiply the coefficients (numbers) together: $-5 \times -2$.

Apply the product rule for exponents to the $x$ terms: $x^{3} \times x^{-11} = x^{3 + (-11)}$.

Apply the product rule for exponents to the $y$ terms: $y^{2} \times y^{-2} = y^{2 + (-2)}$.

Combine the results from the coefficient multiplication and the simplified variables to write the final simplified expression.

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Laws of Exponents

The laws of exponents govern how to simplify expressions involving powers. Key rules include multiplying powers with the same base by adding their exponents, and handling negative exponents by rewriting them as reciprocals. These rules allow simplification of expressions like x^a * x^b = x^(a+b).

Multiplication of Algebraic Expressions

When multiplying algebraic expressions, multiply the coefficients (numerical parts) separately from the variables. For variables with exponents, apply the laws of exponents. This process helps combine terms systematically to simplify the overall expression.

Handling Negative Exponents

Negative exponents indicate the reciprocal of the base raised to the positive exponent, e.g., x^(-n) = 1/x^n. Understanding this allows rewriting terms with negative exponents into fractions or adjusting exponents during multiplication to simplify expressions correctly.

Simplify each exponential expression: (−5x3y2)(−2x−11y−2)(-5x^3y^2)(-2x^{-11}y^{-2})