Problem 24 - Algebra A Combined Function

Question

Solve. $$ 64^{x}=16 $$

Step-by-Step Solution

Answer

\(x = \frac{2}{3}\)

Step 1: Express Both Sides with the Same Base

First, recognize that both 64 and 16 can be written as powers of 4. Rewrite the equation using base 4. 64 can be expressed as \( 4^3 \) because \( 4^3 = 64 \).16 can be expressed as \( 4^2 \) because \( 4^2 = 16 \). Thus, rewrite the equation as: \[(4^3)^x = 4^2 \]

Step 2: Simplify the Left Side

Apply the power of a power property, which states that \( (a^m)^n = a^{m imes n} \). Thus, \((4^3)^x = 4^{3x}\).The equation now is: \[ 4^{3x} = 4^2 \]

Step 3: Set the Exponents Equal

Since the bases are the same (base 4), we can set the exponents equal to each other. This gives us the equation: \[3x = 2 \]

Step 4: Solve for x

To solve for \(x\), divide both sides of the equation by 3:\[ x = \frac{2}{3} \]

Key Concepts

Power of a Power Property

The power of a power property is a rule in exponentiation that is incredibly useful when dealing with exponential expressions. It allows you to simplify expressions when one term with an exponent is raised to another exponent. In simple terms, if you have

• \((a^m)^n\)

The property tells you to multiply the exponents:

• \(a^{m \times n}\).

This means that each time you encounter a situation where a combination of exponents exists within parentheses, you can simplify it quickly by applying this rule. This property is fundamental as it turns complex chains of exponents into simple multiplications, streamlining the expression significantly to make further calculations easier. Not only does it help in mathematics, but mastering it can also improve understanding in scientific contexts where exponents frequently appear.

Simplifying Expressions

Simplifying expressions is an essential skill in algebra. It involves rewriting expressions in a simpler or more efficient way. In the exercise given, the simplification begins by expressing numbers like 64 and 16 with the same base. Here:

• 64 is simplified to \(4^3\)
• 16 is simplified to \(4^2\)

Rewriting the numbers with a common base helps expose the deeper structure of the equation, allowing properties of exponents (like the power of a power property) to be used effectively. Once the expression is simplified, the problem becomes much easier to tackle, as seen with the transformed equation \((4^3)^x = 4^2\). This not only aids in solving complex equations step-by-step but also reduces error and enhances comprehension of underlying mathematical principles.

Solving Equations

Solving equations is the process of finding the variable that satisfies an equality. In exponential equations, like the one provided, this process often involves using the properties of exponents. Once rewritten as:

• \(4^{3x} = 4^2\)

The art of solving such equations lies in identifying that, with identical bases, you can equate the exponents. Thus:

• set \(3x = 2\)

And solve by isolating the variable. In our example, divide both sides by 3 to find \(x = \frac{2}{3}\). This technique of setting bases equal is cornerstone in solving exponential equations. Mastery in this not only aids in efficiently handling equations of power but also in grasping the concept of exponential growth and decay encountered in various scientific and real-world scenarios.