Problem 51 - College Algebra Essentials

Question

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

Step-by-Step Solution

Answer

-5a^{11}b

Step 1: Dealing with the Numerical Part

First, divide the numerical coefficients 25 and -5. Remember that dividing by a negative number will change the sign, hence \( \frac{25}{-5} = -5 \). The result is negative since dividing a positive number by a negative number always results in a negative quotient.

Step 2: Simplifying Exponents of a

Next deal with \( a^{13} \) and \( a^{2} \). Since these are terms with the same base being divided, subtract the exponents, i.e. \( a^{13} \div a^{2} = a^{13-2} = a^{11} \).

Step 3: Simplifying Exponents of b

Proceed similarly with \( b^{4} \) and \( b^{3} \), i.e. \( b^{4} \div b^{3} = b^{4-3} = b^{1} = b \), since any number raised to the power of 1 is itself.

Step 4: Final Result

Lastly, combine the results from steps 1-3 to get the final simplified expression: -5a^{11}b.