Problem 116 - Elementary Algebra

Question

Simplify the following problems. $$ \frac{(2 x-1)^{13}(2 x+5)^{5}}{(2 x-1)^{10}(2 x+5)} $$

Step-by-Step Solution

Answer

Question: Simplify the given expression: $$ \frac{(2 x-1)^{13}(2 x+5)^{5}}{(2 x-1)^{10}(2 x+5)} $$ Answer: $$ (2x-1)^3(2x+5)^4 $$

Step 1: Identify the common factors

Observe that \((2x-1)^{10}\) and \((2x+5)\) are common factors in both the numerator and the denominator.

Step 2: Apply the properties of exponents

Using the property of exponents that states \(a^{m} \div a^{n} = a^{m-n}\), we can cancel the common factors. Here, \(a\) represents the base and \(m\) and \(n\) are different exponents.

Step 3: Simplify the expression

Now, we can apply the exponents property and simplify the expression: $$ \frac{(2 x-1)^{13}(2 x+5)^{5}}{(2 x-1)^{10}(2 x+5)} = \frac{(2x-1)^{13-10}(2x+5)^{5-1}}{(2x-1)^{10}(2x+5)} $$

Step 4: Complete the calculation

Now, we just have to compute the new exponents and write the final form of the expression: $$ \frac{(2x-1)^{3}(2x+5)^{4}}{(2x-1)^{10}(2x+5)} = \frac{(2x-1)^3(2x+5)^4}{(2x-1)^{10}(2x+5)} $$ The simplified expression is: $$ (2x-1)^3(2x+5)^4 $$

Key Concepts

Properties of Exponents

Understanding the properties of exponents is crucial in simplifying expressions like the one in the given exercise. Exponents indicate how many times a number, known as the base, is multiplied by itself.
One of the primary exponent properties used in this exercise is the division property: \(a^m \div a^n = a^{m-n}\). This property allows us to subtract the exponent in the denominator from the exponent in the numerator, effectively "canceling out" common terms.
For instance, in the expression \((2x-1)^{13} \div (2x-1)^{10}\), the bases are the same. By applying the division property, we subtract the exponents:

• Your new exponent becomes \(13 - 10 = 3\), simplifying \((2x-1)^{13}\) to \((2x-1)^3\).

This concept helps streamline complex expressions and make equations more manageable.

Common Factors

Common factors refer to elements that appear in both the numerator and the denominator of a fraction.
Identifying common factors is the first step when simplifying algebraic expressions because they can be "canceled" or removed from both parts of the fraction, reducing its complexity.

• In our problem, the terms \((2x-1)^{10}\) and \((2x+5)\) are common factors in both the numerator and the denominator.
• By identifying and removing these common factors, the expression becomes much simpler.

Always look for common factors before moving into deeper simplification, as this can drastically reduce the difficulty of the problem.

Algebraic Simplification

Algebraic simplification involves using mathematical operations and properties to rewrite expressions in a simpler form.
This process makes it easier to work with equations and inequalities in algebra.
It involves:

• Identifying common factors.
• Applying the properties of exponents and other algebraic rules.
• Performing operations such as addition, subtraction, multiplication, and division where applicable.

In the given exercise, simplification was achieved by dividing both the numerator and the denominator by their common factors, then adjusting the exponents using the properties of exponents.
After reducing the expression, the final result was \((2x-1)^3(2x+5)^4\). This type of simplification is essential because it allows us to convey complex mathematical ideas in a more straightforward way, making them easier to understand and solve.