Simplify each exponential expression in Exercises 23–64. $\frac{25a^{13}{b}^4}{-5a^2{b}^3}$

Verified step by step guidance

1. Identify the given expression: \(\frac{25a^{13}b^{4}}{-5a^{2}b^{3}}\).
2. Separate the coefficients and the variables to simplify each part individually: \(\frac{25}{-5} \times \frac{a^{13}}{a^{2}} \times \frac{b^{4}}{b^{3}}\).
3. Simplify the coefficients: \(\frac{25}{-5} = -5\).
4. Apply the quotient rule for exponents to the variables: \(\frac{a^{13}}{a^{2}} = a^{13-2} = a^{11}\) and \(\frac{b^{4}}{b^{3}} = b^{4-3} = b^{1}\).
5. Combine the simplified parts to write the final simplified expression: \(-5a^{11}b\).

Key Concepts

Laws of Exponents

The laws of exponents govern how to simplify expressions involving powers. Key rules include dividing powers with the same base by subtracting exponents, multiplying powers by adding exponents, and handling negative exponents by taking reciprocals. These rules allow simplification of expressions like a^13 / a^2 = a^(13-2) = a^11.

Simplifying Algebraic Fractions

Simplifying algebraic fractions involves reducing the numerator and denominator by canceling common factors. This includes factoring expressions and applying exponent rules to variables. The goal is to write the expression in its simplest form, making it easier to interpret or use in further calculations.

Handling Negative Signs in Fractions

When simplifying fractions, the negative sign can be placed in front of the entire fraction, in the numerator, or in the denominator. Understanding how to correctly manage the negative sign ensures the simplified expression accurately reflects the original value, such as writing -5a^2 b^3 in the denominator as a negative fraction.