Problem 51 - College Algebra

Question

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

Step-by-Step Solution

Answer

The simplified expression is \(-5 a^{11} b\).

Step 1: Simplify Coefficients

Calculate the result of \(\frac{25}{-5}\). The result will be the coefficient of the simplified expression.

Step 2: Simplify Terms with Base \(a\)

Apply the exponential laws on the terms \(a^{13}\) and \(a^{2}\). The law states that \((a^{m})/(a^{n})=a^{m-n}\), subtract the exponent of the denominator from the exponent of the numerator. You should get \(a^{11}\).

Step 3: Simplify Terms with Base \(b\)

Lastly, apply the exponential laws on the terms \(b^{4}\) and \(b^{3}\). The law states that \((b^{m})/(b^{n})=b^{m-n}\), subtract the exponent of the denominator from the exponent of the numerator. You should get \(b^{1}\) or simply \(b\).

Key Concepts

Simplifying Coefficients

When simplifying exponential expressions, the first step is often to simplify the coefficients. Coefficients are the numerical parts of the terms that are not attached to any variable or exponent. In the expression \( \frac{25 a^{13} b^{4}}{-5 a^{2} b^{3}} \), the coefficients are 25 and -5. To simplify them, you perform the division \( \frac{25}{-5} \) which results in -5.

This means the simplified expression will start with a coefficient of -5 in the numerator. Simplifying coefficients is a crucial initial step, as it sets the stage for simplifying the rest of the expression that involves variables and their exponents.

Exponential Laws

Exponential laws or rules are fundamental principles used to manipulate and simplify expressions with exponents. One key rule to remember is:

• If you have the same base and you’re dividing, subtract the exponents. For example,
  • \( \frac{a^{m}}{a^{n}} = a^{m-n} \)
• This rule applies to both numeric and variable base terms.

In our expression, after simplifying the coefficients, we have \( a^{13} \) divided by \( a^{2} \). Using the exponential law, you subtract the exponent of 2 from 13, giving you \( a^{11} \).

Applying exponential laws is vital in maintaining the integrity of the expression while making it simpler and more compact.

Base Terms Simplification

Base terms simplification involves making the variables in an expression easier to work with by applying the exponential laws. After dealing with coefficients and starting with the expression \( -5a^{11}b^{4} \), we now focus on the variable \( b \). Apply the rule \( \frac{b^{4}}{b^{3}} = b^{4-3} = b^{1} \), which can simply be written as \( b \).

The simplified form of the entire expression, after applying the base terms simplification, is \( -5a^{11}b \). Focusing on each base separately helps in understanding and methodically simplifying rather than approaching the problem in a complex lump sum.