Problem 26 - Algebra 2 and Trigonometry

Question

Solve each equation and check. \(9^{x-1}=27^{x}\)

Step-by-Step Solution

Answer

The solution is \(x = -2\).

Step 1: Expressing in terms of base 3

The first step is to express both sides of the equation in terms of the same base. Notice that 9 and 27 are both powers of 3. Specifically, 9 can be written as \(3^2\) and 27 as \(3^3\). Substitute these into the equation: \((3^2)^{x-1} = (3^3)^x\).

Step 2: Applying the power of a power rule

Next, apply the power of a power rule, which states that \((a^m)^n = a^{mn}\). So we have: \(3^{2(x-1)} = 3^{3x}\). Simplifying the exponents on both sides gives \(3^{2x-2} = 3^{3x}\).

Step 3: Equating the exponents

Since the bases are the same (both are base 3), we can set the exponents equal to each other: \(2x-2 = 3x\).

Step 4: Solving for x

Rearrange the equation \(2x - 2 = 3x\) to solve for \(x\). Subtract \(2x\) from both sides to get \(-2 = x\).

Step 5: Checking the solution

Substitute \(x = -2\) back into the original equation to verify the solution: \(9^{-3} = 27^{-2}\). Recalculate both sides: \((3^2)^{-3} = (3^{-6})\) and \((3^3)^{-2} = (3^{-6})\). Since both sides are equal, \(x = -2\) is indeed the correct solution.

Key Concepts

Power of a Power Rule

The 'power of a power rule' is a fundamental principle used when working with exponents. This rule helps simplify expressions where an exponent is raised to another exponent. The formal definition is:

• \((a^m)^n = a^{mn}\)

To understand how it works, consider the exercise where we have expressions like

• \((3^2)^{x-1}\)
• \((3^3)^x\)

The power of a power rule allows us to multiply the exponents, simplifying the expressions to:

• \(3^{2(x-1)}\)
• \(3^{3x}\)

Simplification makes solving exponential equations more straightforward by reducing the complexity of the expressions.

Exponent Properties

Understanding exponent properties is crucial when dealing with equations involving exponents. One key property is that exponents can be combined and manipulated using a set of rules, which include:

• Addition or subtraction: \(a^m \times a^n = a^{m+n}\) and \(\frac{a^m}{a^n} = a^{m-n}\)
• Multiplying exponents using the power rule: \((a^m)^n = a^{mn}\)
• Any number raised to zero is one: \(a^0 = 1\)

In our original exercise, the expression involves transforming each side of the equation into exponents with the same base. By doing so, we simplify and align the terms, making it easier to compare and solve the equation. Knowing these properties is integral to simplifying complex exponential expressions efficiently.

Exponential Equations

Exponential equations often involve unknown variables in the exponent. Solving them involves using properties of exponents to align the bases and exponents, making the equation simpler to tackle. Consider the equation from our exercise:

• \(9^{x-1} = 27^x\)

This can be solved by expressing both sides as a power of a common base, such as 3, leading to

• \((3^2)^{x-1} = (3^3)^x\)

Once the bases are consistent, the next step is straightforward: set the exponents equal to each other (since \(a^m = a^n\), then \(m = n\) when \(a > 0\)) and solve for the variable:

• \(2x - 2 = 3x\)
• \(x = -2\)

With exponential equations, starting with a common base simplifies finding solutions.

Checking Solutions

After solving an exponential equation, it's essential to check the solution for accuracy. Verifying results helps confirm that no errors occurred during the solving process. For the equation \(9^{x-1} = 27^x\), we found \(x = -2\). To check:

• Substitute \(x = -2\) back into the original equation:
• \(9^{-3} = 27^{-2}\)
• Express both sides with a common base (3):
• \((3^2)^{-3} = (3^3)^{-2}\)
• Both sides compute to \(3^{-6}\)

Since both expressions are equal, our solution is indeed correct. Checking solutions gives confidence that the value for \(x\) satisfies the original equation, and it's a good habit to develop.