Problem 49 - Beginning and Intermediate Algebra

Question

Simplify. $$\left(\frac{24 a^{10} b^{-8} c^{7}}{12 a^{6} b^{-3} c^{5}}\right)^{-5}$$

Step-by-Step Solution

Answer

The simplified expression is \(\frac{b^{25}}{32a^{20}c^{10}}\).

Step 1: Simplify the constants

First, let's simplify the constant part of the fraction. We have 24/12, which simplifies to 2.

Step 2: Simplify the exponents

For each variable, we will apply the exponent rules. First, let's address the exponent -5. We know that \(a^{-n} = \frac{1}{a^n}\), which means we can rewrite the expression as follows: \[ \frac{1}{\left(\frac{24 a^{10} b^{-8} c^{7}}{12 a^{6} b^{-3} c^{5}}\right)^{5}} \] Now, let's apply the exponent rules for the variables a, b, and c. Since \(a^{n} / a^{m} = a^{n-m}\), and using a similar rule for b and c, we can simplify the expression further: \[ \frac{1}{\left(2\cdot a^{10-6} b^{-8-(-3)} c^{7-5}\right)^5} \] which simplifies to: \[ \frac{1}{\left(2\cdot a^{4} b^{-5} c^{2}\right)^5} \]

Step 3: Apply the power of 5 to each term

Now, we will apply the power of 5 to each term inside the parentheses. Using the rule \(a^{n\cdot m} = a^{n\times m}\), the expression becomes: \[ \frac{1}{(2^5\cdot a^{4\times 5} b^{-5\times 5} c^{2\times 5})} \] which simplifies to: \[ \frac{1}{(32\cdot a^{20} b^{-25} c^{10})} \]

Step 4: Rewrite negative exponents as fractions

Finally, let's rewrite the term with the negative exponent as a fraction (using the rule \(a^{-n} = \frac{1}{a^n}\)): \[ \frac{1}{32\cdot a^{20} b^{-25} c^{10}} = \frac{1}{32\cdot a^{20} \cdot \frac{1}{b^{25}} \cdot c^{10}} \]

Step 5: Final simplified expression

Now, putting everything together, we have the simplified expression: \[ \frac{b^{25}}{32 a^{20} c^{10}} \] So, the final simplified expression is \(\frac{b^{25}}{32a^{20}c^{10}}\).

Key Concepts

Simplifying Algebraic Fractions

Simplifying algebraic fractions involves reducing fractions that contain variables. The aim is to simplify expressions and combine like terms to make them easier to work with. In algebraic fractions, both the numerator and the denominator consist of algebraic expressions. The simplifying process can be accomplished by dividing the coefficients, which are the numbers in front of the variables, and applying the rules of exponents to the variables themselves.
For example, in the fraction \( \frac{24}{12} \), dividing the coefficients gives us 2, simplifying the fraction to \( \frac{2}{1} \). It's just basic division! Even though it looks more complex in algebra, the same principles apply. This simplification helps in making complex problems easier to manage and solve.

Negative Exponents

Negative exponents can seem tricky at first, but they are just a different way of expressing fractions. When we see a negative exponent, it tells us to take the reciprocal of the base and change the sign of the exponent. For instance, \( a^{-n} \) is transformed into \( \frac{1}{a^n} \).
As seen in the original solution, this concept is crucial when simplifying algebraic fractions, especially when facing expressions like \( b^{-5} \). We rewrite it as \( \frac{1}{b^5} \) to make calculation simpler. Remember, a negative exponent does not mean the number itself is negative, it merely indicates an inverse process.

Power of a Power Rule

The power of a power rule is an essential exponent rule used to simplify expressions where a power is raised to another power. Mathematically, this rule is expressed as \( (a^m)^n = a^{m \times n} \). This means you multiply the exponents when raising a power to another power.
Let's apply this to the expression \( (a^4)^5 \): According to the power of a power rule, this becomes \( a^{4 \times 5} \), which simplifies to \( a^{20} \).
This rule is a time-saver when you face complex expressions, as it allows you to condense them by multiplying exponents instead of expanding and multiplying terms individually.

Fractional Exponents

Fractional exponents represent roots in algebra. A fractional exponent like \( a^{\frac{m}{n}} \) means the \( n \)-th root of \( a^m \). For instance, \( a^{\frac{1}{2}} \) is the square root of \( a \), and \( a^{\frac{1}{3}} \) represents the cube root.
It's vital to understand this relationship because it allows for flexibility when manipulating expressions. While often seen as intimidating, fractional exponents follow the same laws as regular exponents, which provides a consistent framework for algebraic simplification.
In practice, using fractional exponents can drastically simplify complex root problems through application of exponent rules, making it a powerful tool in advanced mathematics.