Problem 13 - Algebra for College Students

Question

Solve each of the equations. $$ 16^{x}=64 \quad\left\\{\frac{3}{2}\right\\} $$

Step-by-Step Solution

Answer

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

Step 1: Identify Common Base

First, we notice both numbers, 16 and 64, can be expressed as powers of 2. We know that \(16 = 2^4\) and \(64 = 2^6\). Therefore, we can rewrite the equation.

Step 2: Rewrite the Equation in Terms of Powers of 2

Rewrite the original equation \(16^x=64\) as \((2^4)^x = 2^6\).

Step 3: Apply the Power of a Power Rule

Apply the rule \((a^m)^n = a^{m \cdot n}\) to the left side of the equation: \((2^4)^x = 2^{4x}\). The equation now becomes \(2^{4x} = 2^6\).

Step 4: Equate the Exponents

Since the bases are the same (both are 2), we can equate the exponents: \(4x = 6\).

Step 5: Solve for x

Divide both sides of the equation by 4 to solve for \(x\): \(x = \frac{6}{4} = \frac{3}{2}\).

Key Concepts

Common Base Identification

Before we can dive into solving exponential equations, it's crucial to understand how identifying a common base can simplify the process. Exponential expressions often involve numbers that can be broken down into powers of smaller, often prime, numbers. For instance, in the equation above — where you have two different bases (16 and 64) — you'll find that both numbers can be expressed as powers of the base number 2.

• 16 can be expressed as \(2^4\).
• 64 can be reflected as \(2^6\).

By doing this, we can transform a seemingly complex equation into a much more manageable one by rewriting all terms with a common base. Once you have a unified base, comparing the exponents becomes straightforward and actionable.

Power of a Power Rule

When working with exponential expressions, you'll often come across terms nested within each other, such as \((a^m)^n\). This is where the "Power of a Power" rule is immensely useful. This rule states that when you raise a power to another power, you can multiply the exponents.
For example, given an expression \((2^4)^x\), applying the rule transforms it into \(2^{4x}\). This simplification allows us to work directly with the exponents rather than dealing with nested powers.
It's an incredibly handy tool for simplifying equations, especially in tests where time is of the essence.

Equating Exponents

Equating the exponents is a pivotal step when solving exponential equations. Once you have transformed both sides of your equation to have the same base, you can set their exponents equal to each other.
In our example, transforming \(16^x = 64\) gives us \(2^{4x} = 2^6\). Since the base is now the same on both sides, namely the base 2, the exponents must also be equal for the equation to hold true.

• Equation becomes \(4x = 6\)

This step drastically simplifies the problem, as it reduces a complex exponential equation to a simple linear equation that we can effortlessly solve.

Solving Linear Equations

Having equated the exponents, you're now left with a linear equation, such as \(4x = 6\). Solving linear equations is generally straightforward:

• Use inverse operations to isolate the variable.
• In this case, divide both sides by 4 to get \(x = \frac{6}{4} = \frac{3}{2}\).

Linear equations are simpler to deal with since they involve direct calculations. Here, only basic arithmetic is required to find the value of \(x\). It’s always a pleasure to see a complex-looking equation distilled into such an elegant resolution, providing both accuracy and simplicity.