Problem 51 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

The simplified form of the given exponential expression is \(-5a^{11}b\).

Step 1: Separate Coefficients and Variables

Start by separating the coefficients (numbers) from the variables (letters with exponents): \( \frac{25}{-5} * \frac{a^{13}}{a^{2}} * \frac{b^{4}}{b^{3}} \)

Step 2: Simplify Coefficients and Exponents

Then, simplify the separated parts: \(-5 * a^{13-2} * b^{4-3}\)

Step 3: Simplify Further

Further simplification gives: \(-5 * a^{11} * b\)

Step 4: Combine to Final Form

Finally, combine the separated parts into the final simplified form: \(-5a^{11}b\)

Key Concepts

Exponential Expression

The term 'exponential expression' refers to a mathematical notation that conveys repeated multiplication of a base number raised to an exponent. In the context of our problem, the expression \(25 a^{13} b^{4}\) features both a constant (25) and variables (\(a\) and \(b\)) raised to exponents (13 and 4, respectively).

To simplify an exponential expression, it's crucial to understand that an exponent indicates how many times a base is multiplied by itself. For example, \(a^{13}\) means \(a\) is multiplied by itself 13 times.

Laws of Exponents

When dealing with exponential expressions, 'laws of exponents' are critical rules for carrying out algebraic simplification. They allow us to manipulate exponents in various ways to simplify expressions.

• For division, \(a^m/a^n = a^{m-n}\) simplifies to the base raised to the difference of the exponents.
• When multiplying like bases, you add exponents: \(a^m * a^n = a^{m+n}\).
• The power of a product rule \( (ab)^n = a^n * b^n\) expresses that each base in a product is raised to the exponent.

Using these rules, we can tackle complex expressions systematically, such as the given problem, converting it into a more straightforward form.

Algebraic Simplification

Algebraic simplification involves combining like terms and reducing expressions to their simplest forms. This process frequently involves using the laws of exponents. For instance, in our exercise, we reduced \(a^{13}/a^{2}\) to \(a^{11}\) by subtracting the exponent of the denominator from the exponent of the numerator, as per the laws of exponents.

The goal is to minimize the complexity of the expression while keeping its value unchanged. This clarity can often reveal deeper insights into the problem and lead to easier computation or further manipulation.

Coefficient Simplification

Simplifying coefficients refers to reducing the numerical part of an algebraic expression to its simplest form. In our example, the coefficient simplification is accomplished by dividing the numerical coefficient of the numerator (25) by that of the denominator (-5).

This calculation results in \(25/-5 = -5\), which is then combined with the variables that have been simplified using their respective exponents. Coefficient simplification often precedes or is done simultaneously with the simplification of the variables' exponents to achieve a fully simplified result.