Problem 54 - A Graphical Approach to College Algebra

Question

Solve each equation. $$(\sqrt[4]{3})^{-x}=\left(\frac{1}{3}\right)^{x-1}$$

Step-by-Step Solution

Answer

The solution is \(x = \frac{4}{3}\).

Step 1: Rewrite the Equation

Start by rewriting the given equation to make it easier to work with. The given equation is \((\sqrt[4]{3})^{-x} = \left(\frac{1}{3}\right)^{x-1}\). Notice that \(\sqrt[4]{3}\) can be rewritten as \(3^{1/4}\). Substitute this into the equation: \((3^{1/4})^{-x} = \left(\frac{1}{3}\right)^{x-1}\).

Step 2: Apply Exponent Rules

Use exponent rules to simplify \((3^{1/4})^{-x}\) to \(3^{-x/4}\). The equation now becomes \(3^{-x/4} = \left(\frac{1}{3}\right)^{x-1}\). Also rewrite \(\left(\frac{1}{3}\right)^{x-1}\) as \(3^{-(x-1)}\). We now have \(3^{-x/4} = 3^{-(x-1)}\).

Step 3: Set the Exponents Equal

Since the bases on both sides of the equation are the same (and non-zero), the exponents must be equal for the equation to hold. Therefore, set the exponents equal: \(-\frac{x}{4} = -(x-1)\).

Step 4: Solve for x

To solve for \(x\), first eliminate the negatives:\(\frac{x}{4} = x - 1\).Next, multiply every term by 4 to clear the fraction:\(x = 4x - 4\).Rearranging the equation gives:\(0 = 3x - 4\).Solving for \(x\), add 4 to both sides and divide by 3:\(3x = 4\) gives \(x = \frac{4}{3}\).

Key Concepts

Exponents and Powers

Exponents and powers are fundamental concepts in mathematics that express how many times a number, the base, is multiplied by itself. These concepts simplify calculations involving large numbers by using a concise notation. For example, \( a^n \) represents a base \( a \) raised to an exponent \( n \). This means \( a \) is multiplied by itself \( n \) times.

When handling exponents, several key rules can greatly assist:

• Product Rule: \( a^m \times a^n = a^{m+n} \). This rule states that when multiplying two powers with the same base, you add the exponents together.
• Quotient Rule: \( \frac{a^m}{a^n} = a^{m-n} \). This explains how to handle division with the same base—subtract the exponents.
• Power of a Power Rule: \( (a^m)^n = a^{m \cdot n} \). This rule describes how to simplify a power raised to another power.
• Negative Exponent Rule: \( a^{-n} = \frac{1}{a^n} \). This shows you how to interpret negative exponents by creating a reciprocal.

Understanding these rules is crucial because they allow you to manipulate and simplify expressions effectively, making complex calculations manageable. Applying these rules was key in the original problem to transform and simplify terms.

Algebraic Equations

In mathematics, algebraic equations are statements where two expressions are set equal to each other, typically containing variables. The main goal in solving these equations is to find the values of variables that make the equation true.

An algebraic equation can take various forms. Some linear equations involve simple terms with no exponents, while others, like polynomials, can have multiple terms and higher degree powers. Despite their complexity, the structure follows these basic steps:

• Identify and isolate the variable you wish to solve for.
• Simplify the equation by distributing constants and combining like terms if necessary.
• Use inverse operations to solve for the variable.
• Check your solution by substituting it back into the original equation to ensure both sides remain equal.

In the provided solution, identifying the equality of exponents meant aligning equations to have the same exponential base, which simplified the problem to an algebraic equation with one variable.

Solving Equations

Solving equations involves finding the value(s) of variable(s) that balance an equation. When dealing with equations involving exponents and powers, this task often involves multiple steps. Each step requires a keen understanding of how to manipulate terms to isolate the variable of interest.

Consider the given solution involves these essential steps:

• First, rewrite complex expressions into manageable forms using algebraic rules, such as changing \( \sqrt[4]{3} \) to \( 3^{1/4} \).
• Next, leverage exponential rules to equate bases. In our case, \( (\frac{1}{3})^{x-1} \) was rewritten as \( 3^{-(x-1)} \).
• Once the bases are equal, set the exponents equal to solve the simpler equation resulting from the alignment of bases.

Solving such equations often means working systematically. After setting exponents equal, you solve the linear equation step-by-step to arrive at \( x = \frac{4}{3} \).

Understanding these steps can enhance your problem-solving toolkit, making the handling of equations with exponents a bit more straightforward.