Problem 10 - Algebra and Trigonometry with Analytic Geometry

Question

Solve the equation. $$9^{2 x} \cdot\left(\frac{1}{3}\right)^{x+2}=27 \cdot\left(3^{x}\right)^{-2}$$

Step-by-Step Solution

Answer

The solution is \(x = 1\).

Step 1: Understand the problem

The given equation is \(9^{2x} \cdot \left(\frac{1}{3}\right)^{x+2} = 27 \cdot \left(3^{x}\right)^{-2}\). Our task is to find the value of \(x\). To make the problem easier, we'll rewrite the terms with the same base.

Step 2: Rewrite terms with a common base

Notice that 9, 27, and \(\frac{1}{3}\) can be rewritten with base 3.- \(9 = 3^2\) so \(9^{2x} = (3^2)^{2x} = 3^{4x}\)- \(\frac{1}{3} = 3^{-1}\) so \(\left(\frac{1}{3}\right)^{x+2} = (3^{-1})^{x+2} = 3^{-(x+2)}\)- \(27 = 3^3\)- \(\left(3^{x}\right)^{-2} = 3^{-2x}\)After rewriting, we get:\[3^{4x} \cdot 3^{-(x+2)} = 3^3 \cdot 3^{-2x}\]

Step 3: Simplify using properties of exponents

Use the property of exponents that states \(a^m \cdot a^n = a^{m+n}\). Combine the powers of 3 on the left side:\[3^{4x - (x+2)} = 3^{3 - 2x}\]This simplifies to:\[3^{3x - 2} = 3^{3 - 2x}\]

Step 4: Equate the exponents

Since the bases are the same, we can equate the exponents:\[3x - 2 = 3 - 2x\]

Step 5: Solve the equation for x

Rearrange the equation to find \(x\):\[3x + 2x = 3 + 2\]\[5x = 5\]Divide both sides by 5:\[x = 1\]

Key Concepts

Rewriting Bases

The foundation of solving exponential equations often involves rewriting numbers to have a common base. This is key because it simplifies the equation, allowing direct comparison of exponents. In our exercise, the numbers 9, 27, and \(\frac{1}{3}\) can all be expressed as powers of 3.

• 9 is rewritten as \(3^2\), which becomes \((3^2)^{2x} = 3^{4x}\).
• \(\frac{1}{3}\) is rewritten as \(3^{-1}\), so \((\frac{1}{3})^{x+2} = (3^{-1})^{x+2} = 3^{-(x+2)}\).
• 27 is \(3^3\).
• Finally, \((3^x)^{-2}\) becomes \(3^{-2x}\).

By rewriting the terms with a common base, we transform the original equation into a simpler form: \(3^{4x} \cdot 3^{-(x+2)} = 3^3 \cdot 3^{-2x}\). This simplification is essential for the next steps.

Properties of Exponents

The next logical step in solving the problem is to simplify the expression using the properties of exponents. These properties are mathematical rules that describe how exponents work. They help you manipulate exponential expressions easily. A fundamental property we use is:

• \(a^m \cdot a^n = a^{m+n}\)

Using this property, we combine the bases on each side:

• On the left side, \(3^{4x} \cdot 3^{-(x+2)}\) simplifies to \(3^{4x - (x+2)} = 3^{3x - 2}\).
• On the right side, \(3^3 \cdot 3^{-2x}\) simplifies to \(3^{3 - 2x}\).

Now, because both sides of the equation have a common base of 3, we can simplify further by focusing on the exponents. This sets the stage for solving the equation easily.

Equating Exponents

Once the bases are the same, you can equate the exponents because if \(a^m = a^n\), then \(m = n\). This is crucial as it translates an exponential equation into a simple linear equation. In our problem, the equation simplifies to:

• \(3x - 2 = 3 - 2x\)

We now solve for \(x\) by:

• Rearranging the terms: \(3x + 2x = 3 + 2\)
• Simplifying it to \(5x = 5\)
• Dividing both sides by 5, which gives us \(x = 1\)

By equating the exponents, a seemingly complex exponential equation is reduced to a straightforward arithmetic calculation, resulting in a clear solution for \(x\). This process not only provides the solution but also reinforces the logical flow from exponential to linear forms.