Problem 523 - Precalculus

Question

Solve \(216^{3 x} \cdot 216^{x}=36^{3 x+2}\) by rewriting each side with a common base.

Step-by-Step Solution

Answer

The solution is \(x = \frac{2}{3}\).

Step 1: Express Each Term Using a Common Base

Both 216 and 36 can be expressed as powers of 6. Notice that \(216 = 6^3\) and \(36 = 6^2\). Let's rewrite the equation \(216^{3x} \cdot 216^x = 36^{3x + 2}\) in terms of base 6. This becomes \((6^3)^{3x} \cdot (6^3)^x = (6^2)^{3x + 2}\).

Step 2: Apply Power to Power Rule

Apply the power to power rule \((a^m)^n = a^{m \cdot n}\) to each term: \[ (6^3)^{3x} = 6^{9x} \text{ and } (6^3)^x = 6^{3x} \text{ on the left side.} \]So the equation becomes: \[ 6^{9x} \cdot 6^{3x} = 6^{2(3x + 2)} \]

Step 3: Simplify Left Side Using Product of Powers Rule

On the left, use the product of powers rule, which states \(a^m \cdot a^n = a^{m+n}\):\[ 6^{9x} \cdot 6^{3x} = 6^{12x} \].The equation simplifies to: \[ 6^{12x} = 6^{6x + 4} \]

Step 4: Set Exponents Equal

Since the bases are the same, set the exponents equal to each other:\[ 12x = 6x + 4 \].

Step 5: Solve for x

Solve the equation \(12x = 6x + 4\) by subtracting \(6x\) from both sides:\[ 12x - 6x = 4 \],which simplifies to:\[ 6x = 4 \].Now divide both sides by 6:\[ x = \frac{4}{6} \],which reduces to:\[ x = \frac{2}{3} \].

Key Concepts

Common Base

The common base method is a powerful tool in solving exponential equations. It involves rewriting different terms of the equation to have the same base number.

• This simplifies the comparison of exponents, enabling straightforward solutions.
• For example, numbers like 216 and 36 can both be expressed as powers of the base number 6.
• In our example, 216 is written as \(6^3\), and 36 as \(6^2\).

By transforming each term into powers of a common base, you convert complex exponential expressions into simpler exponents. This conversion is key to managing and solving exponential equations with ease.
Making numbers appear with a common base is like "speaking the same language" in mathematics; it ensures terms can be directly compared.

Power to Power Rule

The power to power rule is essential when dealing with expressions that involve an exponent being raised to another exponent. It helps simplify such expressions by providing a straightforward way to combine powers.

• Officially, the rule is expressed as: \((a^m)^n = a^{m \cdot n}\).
• This rule implies multiplying the exponents together.

In the given problem, we apply the rule to the expressions \((6^3)^{3x}\) and \((6^3)^x\), which transforms them into \(6^{9x}\) and \(6^{3x}\), respectively.
The power to power rule plays a crucial role in simplifying exponential expressions, making subsequent calculations more manageable.

Product of Powers Rule

The product of powers rule is another foundational concept in working with exponents. It provides a method to combine like bases multiplied together by summing their exponents.

• The rule is presented as \(a^m \cdot a^n = a^{m+n}\).
• This rule is useful when simplifying complex equations.

In our scenario, we applied this rule to combine \(6^{9x}\) and \(6^{3x}\) into a single expression, resulting in \(6^{12x}\).
This simplification allows us to more easily equate the exponents, as the bases are already equivalent. The product of powers rule helps smooth out exponential fractions, making them uniform.

Exponents

Exponents represent the number of times a base is multiplied by itself. Understanding how to manipulate exponents is crucial in solving exponential equations.

• An expression like \(a^n\) means the base \(a\) is multiplied by itself \(n\) times.
• Exponents serve as a shorthand notation for repeated multiplication.

In equations like the one given, exponents form the core part of the expressions. By solving equations with a common base, we reduce the problem to managing exponents directly.
Once simplified, these equations often lead to simpler linear equations involving exponents, which can be solved for variables like \(x\). Understanding exponents, whether through rules for powers or by solving with a common base, is central to mastering exponential equations.