Problem 8 - Algebra 2 and Trigonometry

Question

In \(3-10,\) find the value of \(x\) to the nearest hundredth. $$ \frac{x}{e^{3}}=e^{-2} $$

Step-by-Step Solution

Answer

The value of \(x\) is approximately 2.72.

Step 1: Understand the Original Equation

The equation provided is \( \frac{x}{e^3} = e^{-2} \). This equation shows a fraction where \(x\) is in the numerator and \(e^3\) is in the denominator.

Step 2: Isolate x

To solve for \(x\), we need to get \(x\) by itself on one side of the equation. We can do this by multiplying both sides of the equation by \(e^3\). This gives us \(x = e^{-2} \times e^3\).

Step 3: Simplify Using Exponent Rules

We apply the rule of exponents that states \(a^m \times a^n = a^{m+n}\). Thus we have:\[x = e^{(-2) + 3} = e^{1}\].

Step 4: Calculate Exponential Value

The expression \(e^1\) simply equals \(e\). The approximate numerical value of \(e\) is 2.71828. Therefore, to the nearest hundredth, \(x\) is 2.72.

Key Concepts

Isolation of Variables

In many mathematical equations, determining the value of a specific variable is crucial. This process is referred to as isolation of variables. Essentially, it involves rearranging the equation so the variable of interest stands alone on one side. Consider our original equation: \[ \frac{x}{e^3} = e^{-2} \] To isolate \(x\), we need to eliminate everything else on its side of the equation.

• What's in our way? In this case, \(e^3\) is dividing \(x\).
• How do we remove it? Multiply both sides by \(e^3\), making the equation: \[ x = e^{-2} \times e^3 \]

By applying this technique, we effectively "free" \(x\) from the constraints of division by \(e^3\). With \(x\) isolated, we're set up perfectly to simplify further using exponent rules.

Exponent Rules

Exponent rules are fundamental tools in algebra that simplify expressions where numbers are raised to powers. In our problem, we deal with the equation: \[ x = e^{-2} \times e^3 \] Applying exponent rules here helps streamline the calculation. Key rules include:

• Product of Powers Rule: When multiplying like bases, add the exponents. This gives us: \(e^{-2 + 3}\).
• With this rule in mind, simplify the expression to: \(e^{1}\).

Exponent rules allow us to transform complex expressions into simpler forms, an invaluable skill when solving equations.

Calculating Exponential Values

After simplifying, it’s time to calculate the exponential value. The simplified form of our equation is: \[ x = e^{1} \] Calculating \(e^1\) can be straightforward:

• Recognize that \(e^1\) is simply \(e\), the base of natural logarithms.
• The approximate numerical value of \(e\) is 2.71828, widely used in scientific calculations.

Thus, the value of \(x\) is approximately 2.72 when rounded to the nearest hundredth. Understanding how to compute exponential values is vital, especially because they occur frequently in various scientific contexts.