Problem 36 - Algebra A Combined Function

Question

Solve. $$ \left(\frac{1}{9}\right)^{x}=27^{2-x} $$

Step-by-Step Solution

Answer

The solution is \(x = 6\).

Step 1: Convert to Same Base

The first step is to express both sides of the equation with the same base. Notice that 9 and 27 can be written as powers of 3: - 9 is equal to \(3^2\)- 27 is equal to \(3^3\)Now, rewrite the equation: \[\left(\frac{1}{9}\right)^{x} = 27^{2-x} \Rightarrow \left(3^{-2}\right)^x = (3^3)^{2-x}\].

Step 2: Apply Power of a Power Property

Apply the property of exponents \((a^m)^n = a^{mn}\). This simplifies the equation to\[3^{-2x} = 3^{6-3x}\].

Step 3: Remove the Base

Since the bases are the same, we can set the exponents equal to each other: \[-2x = 6 - 3x\].

Step 4: Solve for x

Add \(3x\) to both sides:\[-2x + 3x = 6\]This simplifies to:\[x = 6\].

Key Concepts

Base Conversion

Base conversion is a crucial technique when working with exponential equations. It involves rewriting numbers so that they share the same base, which allows us to compare and manipulate their exponents more easily. In our exercise, we were given an equation with numbers 9 and 27. By recognizing that both can be expressed as powers of 3 (since 9 is \(3^2\) and 27 is \(3^3\)), we converted the original equation into a form with a common base:

• Convert \(\frac{1}{9}\) to \((3^{-2})\) because \(9 = 3^2\), so \(\frac{1}{9} = 3^{-2}\).
• Convert \(27\) to \(3^3\) as already noted.

Now both sides of the equation use base 3, paving the way for further simplification.

Power of a Power

The power of a power property is one of the basic rules of exponents, which states that when you raise an exponent to another power, you multiply the exponents together. This is beautifully expressed as \((a^m)^n = a^{mn}\). In our example, this rule is applied after converting to a common base:

• On the left side, \((3^{-2})^x = 3^{-2x}\).
• On the right side, \((3^3)^{2-x} = 3^{3(2-x)} = 3^{6-3x}\).

By applying this rule, we simplified the equation into a form where we can easily compare the exponents since they now have the same base of 3.

Properties of Exponents

Understanding the properties of exponents is essential when working with equations of this nature. Here are some key properties as applied in our problem:

• Power of a Product: \((ab)^m = a^m b^m\)
• Power of a Power: \((a^m)^n = a^{mn}\)
• Equal Bases: If \(a^m = a^n\), then \(m = n\)

In the given problem, once the equation is simplified to the form \(3^{-2x} = 3^{6-3x}\), we use the equal bases property to derive that their exponents must be equal. This eliminates the bases altogether and allows us to focus solely on solving the resulting simple equation with the exponents.

Solving Equations

Solving exponential equations often involves a few strategic steps. Upon simplifying our equation so that both sides share a common base, we were left with the simple equation \(-2x = 6 - 3x\), using the property of equal exponents. Solving this linear equation involves straightforward algebraic manipulation:

• Add \(3x\) to both sides: \(-2x + 3x = 6\).
• Simplify the equation: \(x = 6\).

This algebraic step resolves the equation completely, demonstrating that the solution to the original exponential equation is \(x = 6\). By knowing how to convert bases and apply exponent properties, these equations become much easier to tackle.