Problem 89 - College Algebra

Question

Perform the indicated operation or operations. $$ \frac{(5 x-3)^{6}}{(5 x-3)^{4}} $$

Step-by-Step Solution

Answer

The simplified form of the given expression is \((5x - 3)^{2}\).

Step 1: Identify the Base and the Exponents

In this fraction, the base is \(5x - 3\). The exponent in the numerator is 6 and in the denominator, it's 4.

Step 2: Apply the Quotient of Powers Property

The quotient of powers property states that \(a^{m}/a^{n} = a^{m-n}\). Applying this property to our fraction, we subtract the exponent in the denominator from the exponent in the numerator: \((5x - 3)^{6 - 4} = (5x - 3)^{2}\)

Step 3: Simplify the Expression

Now we have \((5x - 3)^{2}\) as the simplified form of the original expression. It cannot be simplified further.

Key Concepts

Quotient of Powers Property

In the world of exponents, the quotient of powers property is like a silver bullet for simplifying expressions that involve division of like bases raised to different powers. This property tells us that when you divide two powers that have the same base, you can subtract the exponent in the denominator from the exponent in the numerator. The result is a new power with the same base, but with the subtracted exponent.

Here's the formula: \(\frac{a^m}{a^n} = a^{m-n}\)

• This means if you have \(a^6/a^4\), you subtract 4 from 6, resulting in \(a^2\).
• The base \(a\) must be the same for this property to work.
• It simplifies expressions by reducing the number of multiplications

Understanding this property is crucial for simplifying expressions efficiently and effectively.

Simplifying Expressions

Simplifying expressions is a key skill in algebra that involves transforming complex algebraic terms into simpler ones. The goal is to make the expression easier to understand and solve without changing its value. In the context of exponents, simplifying often involves applying rules like the quotient of powers property.

For example, take the expression \((5x - 3)^6/(5x - 3)^4\). Using the quotient of powers property, this can be simplified to \((5x - 3)^{6-4} = (5x - 3)^2\). This simplified form is much easier to interpret and can be used for further calculations if needed.

• Always simplify exponents wherever possible to reduce complexity.
• Ensure that like terms and bases are identified prior to simplification.
• Simplification helps make mathematical expressions manageable and solvable.

Algebraic Fractions

Algebraic fractions are fractions in which the numerator, denominator, or both contain algebraic expressions. They behave similarly to numerical fractions but often require algebraic techniques for simplification.

To simplify algebraic fractions:

• Look for common factors in the numerator and denominator.
• Apply exponent rules like the quotient of powers property if exponents are involved.
• Reduce the fraction by cancelling out these factors.

For instance, in \(\frac{(5x - 3)^6}{(5x - 3)^4}\), since the base \((5x - 3)\) is common, you can apply the quotient of powers property to simplify it easily. This process highlights how simplifying algebraic fractions often involves traditional fraction techniques with additional algebraic reasoning.