Problem 295 - Precalculus

Question

Use like bases to solve the exponential equation. \(64 \cdot 4^{3 x}=16\)

Step-by-Step Solution

Answer

The solution is \(x = -\frac{1}{3}\).

Step 1: Express as powers of 2

First, note that 64 and 16 can be expressed as powers of 2. We have:- 64 can be written as \(2^6\).- 16 can be written as \(2^4\).This gives us the equation: \(2^6 \cdot 4^{3x} = 2^4\).

Step 2: Express 4 as a power of 2

Next, recognize that 4 is equal to \(2^2\). Therefore, \(4^{3x}\) can be rewritten as \((2^2)^{3x}\), which simplifies to \(2^{6x}\). So, the equation becomes: \(2^6 \cdot 2^{6x} = 2^4\).

Step 3: Use the property of exponents

Apply the property \(a^m \cdot a^n = a^{m+n}\) to combine the terms on the left-hand side:\(2^{6 + 6x} = 2^4\).

Step 4: Set exponents equal to each other

Since the bases are the same, set the exponents equal to one another:\(6 + 6x = 4\).

Step 5: Solve for x

Subtract 6 from both sides to isolate the term with \(x\):\[6x = 4 - 6\]\[-6x = -2\]Divide both sides by 6:\[x = -\frac{1}{3}\]

Key Concepts

Powers of 2

Understanding powers of 2 is a fundamental concept that helps simplify exponential equations. Numbers like 64, 16, and 4 can all be expressed as powers of 2, making them much easier to work with in equations.

Here's how they break down:

• 64 can be written as \(2^6\), which means 2 multiplied by itself 6 times.
• 16 is expressed as \(2^4\), where 2 is multiplied 4 times.
• Finally, 4 can be represented as \(2^2\).

Recognizing these expressions allows us to convert complex multiplications into a simpler exponential format, which is especially helpful when solving equations involving powers. By expressing numbers as powers of a common base, like 2, we reduce the problem to comparing exponents rather than juggling large numbers.

Property of Exponents

The property of exponents is key to simplifying equations efficiently. One important rule is \(a^m \cdot a^n = a^{m+n}\). This property explains how to handle situations where the same base appears in a multiplication.

Consider an example in the context of our exercise: - Once we express everything as powers of 2, the equation \(2^6 \cdot 2^{6x} = 2^4\) can be simplified by adding the exponents on the left: \[2^{6 + 6x} = 2^4\]This rule allows us to combine terms easily and is extremely useful for simplifying exponential equations with like bases. Understanding and applying properties like this allow us to focus on solving for unknowns without getting entangled in arithmetic of large numbers or multiple terms.

Solving for x

Once we've simplified an exponential equation using powers and properties of exponents, the next step is solving for x. With both sides of the equation having the same base, as in \(2^{6 + 6x} = 2^4\), we can rewrite the equation by setting their exponents equal: - \(6 + 6x = 4\)To find x, isolate the term with x:

• First, subtract 6 from both sides: \(6x = 4 - 6\), simplifying to \(6x = -2\).
• Next, divide each side by 6 to solve for x: \(x = -\frac{1}{3}\).

This process highlights the simplicity afforded by using exponential properties and like bases. By equating exponents directly, we turn what could be a complicated problem into a straightforward arithmetic challenge, making it manageable and less prone to errors.