Problem 27 - Algebra 2 and Trigonometry

Question

What is the value of \(n\) if \(8^{3}=2^{n} ?\)

Step-by-Step Solution

Answer

The value of \(n\) is 9.

Step 1: Understand the Exercise

We are given an equation \(8^3 = 2^n\). Our task is to determine the value of \(n\). Since \(8\) and \(2\) are powers of the same base, we can express them in terms of powers of 2.

Step 2: Express Terms with the Same Base

First, express \(8\) as a power of 2. We know that \(8 = 2^3\). Substitute this into the original equation to get \((2^3)^3 = 2^n\).

Step 3: Apply Power Rule

Apply the power of a power rule, which states \((a^m)^n = a^{m \cdot n}\), to simplify \((2^3)^3\). This gives us \(2^{3 \cdot 3} = 2^n\).

Step 4: Simplify Exponents

Multiply the exponents on the left-hand side: \(3 \times 3 = 9\). The equation now becomes \(2^9 = 2^n\).

Step 5: Set Exponents Equal

Since the bases are the same, set the exponents equal to each other: \(n = 9\). This is because \(2^9\) equals \(2^n\), so \(n\) must be 9.

Key Concepts

Power of a Power Rule

When we have an expression like \( (a^m)^n \), the power of a power rule helps us simplify it. This rule tells us that we can simply multiply the exponents.

So, \( (a^m)^n = a^{m \cdot n} \).

This rule is handy because it allows us to manage large exponents easily. For example, in the equation \((2^3)^3 = 2^n\), using this rule transforms it to \(2^{3\cdot3}\), simplifying to \(2^9\).

Understanding this rule:

• Focus on the inner exponent first (\(a^m\)).
• Then apply the outer exponent (\(n\)).
• Multiply these exponents for the result (\(a^{m\cdot n}\)).

This approach helps in breaking down complex exponential expressions into simpler forms.

Exponential Equation

An exponential equation is an equation in which variables are placed in the exponents. These equations often contain terms with the same base and involve operations like multiplying or dividing powers. A typical example is \(a^x = b^y\), where the objective is to determine the values of \(x\) and \(y\).

Resolving exponential equations involves:

• Ensuring terms have the same base.
• Equating and comparing their exponents.
• Solving for the unknown variables.

In the given problem \(8^3 = 2^n\), notice how we express 8 as a power of 2. This allows us to transform the equation into \(2^9 = 2^n\). Once the bases match, we can set the exponents equal to find \(n\), as the equality of bases determines the equality of exponents when the bases are identical.

Base Conversion

Base conversion is a technique used to express numbers or expressions in terms of a different base. In mathematical problems, particularly those involving powers, converting to a common base can simplify the process of solving equations.

For example, numbers like 8, 16, and 32 can be easily represented as powers of 2, with 8 written as \(2^3\), 16 as \(2^4\), etc. This strategy is useful when dealing with exponential equations.

To effectively convert bases:

• Identify a base that both numbers are powers of.
• Rewrite the numbers in terms of this base.
• Simplify the equation using common exponents.

In our exercise, we converted 8 to \(2^3\), which allowed us to handle the problem with a shared base, quickly leading to the solution.