Problem 22 - Intermediate Algebra

Question

For Problems \(1-34\), solve each equation. $$ 8^{3 x+2}=\frac{1}{16} $$

Step-by-Step Solution

Answer

The solution to the equation is \(x = -\frac{10}{9}\).

Step 1: Identify the Bases

Notice that 8 and 16 can both be expressed as powers of 2. Specifically, 8 is equal to \(2^3\) and 16 is equal to \(2^4\).

Step 2: Rewrite the Equation in Base 2

Convert the given equation, \(8^{3x+2} = \frac{1}{16}\), into base 2. This becomes: \((2^3)^{3x+2} = \frac{1}{2^4}\).

Step 3: Apply Exponent Rules

Use the power of a power rule, \((a^m)^n = a^{m }\), for both sides of the equation: \(2^{3(3x+2)} = 2^{-4}\).

Step 4: Simplify Exponents

Simplify the expression on the left side: \(2^{9x + 6} = 2^{-4}\).

Step 5: Set Exponents Equal

Since the bases are equal, set the exponents equal to each other: \(9x + 6 = -4\).

Step 6: Solve for x

Isolate \(x\) by first subtracting 6 from both sides: \(9x = -10\). Then, divide both sides by 9 to solve for \(x\): \(x = -\frac{10}{9}\).

Key Concepts

Exponent Rules

Understanding exponent rules is crucial when dealing with exponential equations. These rules help us manipulate and simplify expressions involving powers. One important rule is the **power of a power** rule:

• For any base \(a\) and exponents \(m\) and \(n\), \((a^m)^n = a^{m \cdot n}\).

This means when an exponentiated number is itself raised to another power, you multiply the exponents.

Another essential rule is the **negative exponent** rule, which states that any base \(b eq 0\) raised to a negative exponent \(-n\) is equal to the reciprocal of the base raised to the positive exponent:

• \(a^{-n} = \frac{1}{a^n}\).

These rules are foundations in solving exponential equations, enabling us to rewrite expressions in different forms that make solving equations easier.

Base Conversion

Base conversion involves rewriting numbers as powers of a different base. This step simplifies equations when the bases are different, as was in our example where 8 and 16 shared the base of 2.

Recognizing common bases among numbers is the key to base conversion.

• For instance, you should know that 8 can be written as \(2^3\) and 16 as \(2^4\).

Converting to a common base allows us to set the exponents of equal bases directly against each other. With our equation, changing the bases of 8 and 16 to 2 simplified the problem greatly.

After converting, solving equations becomes a matter of matching exponents, capitalizing on the new base to simplify complex expressions into simpler exponent equations.

Solving Equations

Solving exponential equations often involves making the bases on both sides of the equation the same and then equating the exponents. After using base conversion and exponent rules, your job is to

• Set the simplified exponents equal.

In our example, once we converted the bases and applied the exponent rules, we got to the form\[ 2^{9x + 6} = 2^{-4} \].

With the bases the same, solving the equation means simply focusing on the exponents:

• \(9x + 6 = -4\)

From here, basic algebra takes over—subtract the terms appropriately and simplify to find \(x\). Just remember the order of operations: handle exponents, convert bases, and equate before using algebra to solve for the unknown.