Problem 300 - Precalculus

Question

Use like bases to solve the exponential equation. \(\left(\frac{1}{64}\right)^{3 n} \cdot 8=2^{6}\)

Step-by-Step Solution

Answer

\(n = -\frac{1}{6}\).

Step 1: Express all numbers as powers of 2

First, express the numbers involved as powers of 2. \(\frac{1}{64}\) is the same as \(2^{-6}\) because \(64 = 2^6\), and the reciprocal gives us \(2^{-6}\). Also, note that \(8 = 2^3\). Thus, the equation becomes \((2^{-6})^{3n} \cdot 2^3 = 2^6\).

Step 2: Apply the power of a power property

Use the property \((a^m)^n = a^{m \cdot n}\). For \((2^{-6})^{3n}\), apply this property to get \(2^{-18n}\). Now the equation is \(2^{-18n} \cdot 2^3 = 2^6\).

Step 3: Combine the exponents on the left side

We know that \(a^m \cdot a^n = a^{m+n}\). Apply this to \(2^{-18n} \cdot 2^3\), resulting in \(2^{-18n + 3}\). Now the equation is \(2^{-18n + 3} = 2^6\).

Step 4: Set the exponents equal to each other

Since the bases are the same (both are powers of 2), set the exponents equal: \(-18n + 3 = 6\).

Step 5: Solve the linear equation for n

Start by isolating terms involving \(n\): \(-18n + 3 = 6\) -> \(-18n = 6 - 3\) -> \(-18n = 3\). Next, divide both sides by -18: \(n = \frac{3}{-18} = -\frac{1}{6}\).

Key Concepts

Powers of 2

Understanding powers of 2 is central to working with exponential equations. A power of 2 simply means a number that can be expressed as \(2^n\), where \(n\) is any whole number. For example, 8 is a power of 2 because it equals \(2^3\), and 64 equals \(2^6\).
Recognizing these powers is crucial when trying to solve equations like the one in our exercise.

• It allows you to convert numbers into a common base, simplifying your calculations.
• Understanding these helps identify equivalent expressions for multiplication and division.

Knowing these basics lets you successfully solve problems by spotting how different numbers relate through their powers of 2.

Power of a Power Property

The power of a power property is a key concept in manipulating expressions like \((a^m)^n\).
The property tells us to multiply the exponents: \((a^m)^n = a^{m \cdot n}\).
In our equation, we had \((2^{-6})^{3n}\). By applying this property, we simplified this to \(2^{-18n}\).

• It reduces complex exponentiation to simpler terms, making equations easier to work with.
• It allows quick computation of powers without repeated multiplication.

Remember, always multiply the exponents when you see a power raised to another power.

Combining Exponents

Multiplying terms with the same base is made simple with the property that \(a^m \cdot a^n = a^{m+n}\). In our equation, we see this with \(2^{-18n} \cdot 2^3\), which combines into \(2^{-18n + 3}\).
This method depends on the bases being the same and allows for straightforward simplification.

• This technique is essential for consolidating like terms in exponential equations.
• It helps in reducing the complexity of equations by turning multiple terms into a single expression.

By skillfully combining exponents, you can make difficult expressions much easier to tackle.

Linear Equations

Solving the simplified version of an exponential equation often turns into a linear equation problem. Here, setting the exponents equal since the bases are the same led us to \(-18n + 3 = 6\).
Linear equations follow the form \(ax + b = c\) and are solved by isolating the variable.
In our case, we arrived at \(-18n = 3\) after moving terms, and then divided by \(-18\) to solve for \(n\).

• Understanding linear equations is crucial for solving for unknown variables once expressions are equated.
• Isolation of the variable through addition, subtraction, and division is a key step in solving these equations.

Mastering linear equations allows you to conclude solutions efficiently after a series of simplifications.