Problem 76 - Algebra and Trigonometry

Question

Based on material learned earlier in the course. The purpose of these problems is to keep the material fresh in your mind so that you are better prepared for the final exam. Simplify: \(\frac{4 \cdot 3^{x} \cdot \ln 3 \cdot x^{1 / 2}-4 \cdot 3^{x} \cdot \frac{1}{2} \cdot x^{-1 / 2}}{(\sqrt{x})^{2}}\)

Step-by-Step Solution

Answer

4 \cdot 3^{x} \cdot ( \ln 3 - \frac{1}{2x})

Step 1 - Simplify the denominator

Recognize that \[ (\sqrt{x})^{2} = x \] So the denominator simplifies to \[x \]

Step 2 - Simplify the numerator

Factor out common terms in the numerator: \[ 4 \cdot 3^{x} \cdot x^{-1/2} \cdot ( x \cdot \ln 3 - \frac{1}{2} ) \]

Step 3 - Combine numerator and denominator

Combine the simplified numerator and denominator: \[ \frac{4 \cdot 3^{x} \cdot x^{-1/2} ( x \cdot \ln 3 - \frac{1}{2})}{x} \]

Step 4 - Simplify the expression

We can cancel out x in the expression: \[ 4 \cdot 3^{x} \cdot x^{-1} ( x \cdot \ln 3 - \frac{1}{2} ) \] results in \[ 4 \cdot 3^{x} \cdot ( \ln 3 - \frac{1}{2x}) \]

Key Concepts

algebra

Algebra forms the foundation for many branches of mathematics and science. It involves manipulating symbols and expressions to solve equations or simplify terms. In this exercise, the goal is to simplify an algebraic expression.

The expression given is: \[ \frac{4 \times 3^{x} \times \text{ln } 3 \times x^{1 / 2}-4 \times 3^{x} \times \frac{1}{2} \times x^{-1 / 2}}{(\text{√}x)^{2}} \]

The process begins by simplifying both the numerator and the denominator, and then combining them to form a simpler expression. Recognizing patterns and pulling out common factors play a key role in simplifying such expressions.

simplification

Simplification in mathematics means to make an expression easier to understand or work with. The goal is to reduce the expression to its simplest form.

For this problem, we start with simplifying the denominator: \(\text{(\text{√}x)}^{2} = x\).

This allows us to rewrite the denominator as just \(x\).
Next, we focus on the numerator by factoring out common terms. Recognizing \(4 \times 3^{x} \times x^{-1/2}\) as common terms helps us rewrite the expression:

\(4 \times 3^{x} \times x^{-1/2} \times ( x \times \text{ln } 3 - \frac{1}{2})\).

Combining the simplified versions of the numerator and denominator gives:

\(\frac{4 \times 3^{x} \times x^{-1/2} ( x \times \text{ln } 3 - \frac{1}{2})}{x}\).

exponents

Exponents are a way to represent repeated multiplication. In the expression \(3^{x}\), \(3\) is the base and \(x\) is the exponent, meaning \(3\) is multiplied by itself \(x\) times.

Simplifying exponents is crucial for simplifying algebraic expressions. For example, \(x^{1/2}\) (square root of \(x\)) combined with \(x^{-1/2}\) (reciprocal of the square root of \(x\)) simplifies to \(1\).

Understanding the laws of exponents helps in manipulating and simplifying expressions quickly.

In this problem, we relied on those laws to combine and rewrite terms effectively, particularly in how \(x^{-1/2}\) in the numerator works with the \(x\) in the denominator.

logarithms

Logarithms (logs) are the inverse of exponentiation and help solve equations involving exponents. \(\text{ln}\) represents the natural logarithm, which uses \(e\) (approximately 2.718) as its base.

In the simplified expression \(4 \times 3^{x} \times ( \text{ln} 3 - \frac{1}{2x})\), \(\text{ln} 3\) emerges as a key term.

Understanding how to simplify logs and use the properties of logarithms, such as \(\text{ln}(a \times b) = \text{ln} a + \text{ln} b\), plays a pivotal role in algebraic simplification.

Logarithms make it easier to deal with multiplication and division by transforming them into addition and subtraction.