Problem 44 - A Graphical Approach to College Algebra

Question

Solve each equation. $$3^{5-x}=1$$

Step-by-Step Solution

Answer

x = 5.

Step 1: Understand the Equation

The equation given is \( 3^{5-x} = 1 \). We want to find the value of \( x \) that makes this true.

Step 2: Recognize the Property

Recognize that any number to the power of 0 is 1, i.e., \( a^0 = 1 \). Therefore, \( 3^{5-x} = 1 \) implies that \( 3^{5-x} = 3^0 \).

Step 3: Set the Exponents Equal

Since the bases are the same (both are 3), we can set the exponents equal to each other: \( 5-x = 0 \).

Step 4: Solve for x

Solve the equation \( 5-x = 0 \). Add \( x \) to both sides to get \( 5 = x \). Therefore, \( x = 5 \).

Key Concepts

Exponents

Exponents are a fundamental concept in mathematics used to represent repeated multiplication. When you see an expression like \( a^n \), it means \( a \) is multiplied by itself \( n \) times. The number \( a \) is known as the base, and \( n \) is the exponent. Understanding exponents is crucial for simplifying equations and solving problems efficiently.

Here's a simple breakdown of exponents:

• \( a^1 = a \): Any number to the first power is itself.
• \( a^0 = 1 \): Any number, except zero, to the zeroth power equals 1.
• \( a^{-n} = \frac{1}{a^n} \): A negative exponent means the reciprocal of the base raised to the positive exponent.

Relating this to the original problem, understanding that \( 3^0 = 1 \) helps in recognizing what modifies the exponent to solve the equation.

Solving Equations

When solving equations, the goal is to find the value(s) of the unknown variable that satisfy the equation. In an exponential equation like \( 3^{5-x} = 1 \), we employ strategies specific to dealing with exponents. The first step is simplifying the equation or recognizing properties that can aid simplification.

Steps to Solve Exponential Equations:

• Identify equations with a common base: Both sides of the equation, if possible, should be re-written with the same base.
• Use properties of exponents: Leverage properties such as \( a^0 = 1 \) to recognize that matching exponents is useful for simplification.
• Set exponents equal: Once the bases match, equate the exponents and solve the resulting simpler linear equation.

In the given problem, recognizing that \( 3^{5-x} \) should be equal to \( 3^0 \) allows us to equate the exponents directly with a focus on basic algebraic manipulation to find \( x = 5 \).

Properties of Exponents

Properties of exponents are essential tools when solving equations involving exponential expressions. These properties simplify complex problems, allowing easier manipulation and solving. Key properties include the zero-exponent rule, power of a power, and product/exponent rules.

Some important properties include:

• Zero Exponent Rule: \( a^0 = 1 \), which applies even in complex equations where the aim is to neutralize the base effect.
• Product Rule: \( a^m \times a^n = a^{m+n} \) makes multiplying numbers with the same base straightforward.
• Power of a Power: \( (a^m)^n = a^{m \times n} \) simplifies expressions where exponents are layered.
• Equal Bases: If \( a^m = a^n \), then \( m = n \), as seen in solving \( 3^{5-x} = 1 \).

Using these properties, particularly setting equal bases and understanding the zero exponent rule, made it possible to solve the problem: \( 3^{5-x} = 3^0 \), leading to setting \( 5-x = 0 \) and hence solving for \( x = 5 \).