Problem 53 - Intermediate Algebra

Question

Write an explanation of the way you would solve the equation \(\left(2^{x+1}\right)\left(8^{2 x-3}\right)=64\).

Step-by-Step Solution

Answer

The value of \(x\) is 2.

Step 1: Simplify the Equation

First, recognize that the equation can be rewritten in terms of powers of 2, by noting that 8 can be rewritten as a power of 2. We know that \(8 = 2^3\). So we rewrite the equation as \( \left(2^{x+1}\right)\left((2^3)^{2x-3}\right)=64\).

Step 2: Apply the Power of a Power Rule

Use the power of a power rule, \((a^m)^n = a^{m\cdot n}\), to simplify \((2^3)^{2x-3}\) to \(2^{3(2x-3)}\) which equals \(2^{6x-9}\). The equation becomes: \(2^{x+1} \cdot 2^{6x-9} = 64\).

Step 3: Combine Exponents

Apply the rules of exponents, \(a^m \cdot a^n = a^{m+n}\). Thus, combine the exponents on the left: \(2^{x+1+6x-9}\), which simplifies to \(2^{7x-8}\). The equation now reads \(2^{7x-8} = 64\).

Step 4: Express 64 as a Power of 2

Recognize that 64 is a power of 2: \(64 = 2^6\). Rewriting the equation gives us: \(2^{7x-8} = 2^6\).

Step 5: Set Exponents Equal and Solve

Since the bases are equal, we can set the exponents equal to each other: \(7x-8 = 6\). Solve for \(x\) by first adding 8 to both sides to get \(7x = 14\), then divide by 7 to find \(x = 2\).

Key Concepts

Laws of Exponents

One of the fundamental tools in solving exponential equations is understanding the laws of exponents. These laws allow us to manipulate expressions in a way that makes them easier to understand and solve. The power of a power rule,

• The rule states: \((a^m)^n = a^{m \cdot n}\). It simplifies expressions where an exponent is raised to another exponent.
• For example, in the expression \((2^3)^{2x-3}\), we apply this rule to get \(2^{6x-9}\).
• This rule helps in reducing complexity and making equations more straightforward.

Another important rule is the product of powers rule:

• The rule states: \(a^m \cdot a^n = a^{m+n}\).
• This is used to combine like bases, as seen in the solution where \(2^{x+1}\) and \(2^{6x-9}\) become \(2^{7x-8}\).
• Understanding these rules is crucial for efficiently solving problems involving exponents.

Powers of 2

The concept of powers of 2 is often encountered in mathematics, especially in exponential equations. Recognizing powers of 2 can greatly simplify solving these equations.

• Any number that can be expressed as a power of 2 has a form \(2^n\) for some integer \(n\).
• For instance, the number 64 can be expressed as \(2^6\), which is crucial for simplifying the equation at hand.
• In many exercises, walls can be knocked down by converting all numbers in the equation to powers of 2, hence uniformizing the expression for easier handling.

Identifying numbers like 8 or 64 as powers of 2 means:

• 8 becomes \(2^3\), as used in simplifying the original equation.
• This step often serves as the foundational work to make the problem solvable.

Equation Simplification

Simplifying exponential equations is key to solving them effectively. The process involves expressing all terms in the simplest form possible and can sometimes require combining multiple steps.

• Start by expressing all numbers in the equation as powers of a common base, if possible. This was seen when converting 8 to \(2^3\).
• Next, apply the power of a power and product of powers rules to simplify terms into a single exponent.

A critical part of simplification is combining terms with the same base:

• After using the exponent rules, all terms with the base of 2 in the example equation became \(2^{7x-8}\), simplifying the equation to one term with a single exponent.

Once simplified, compare the exponents of the terms:

• In our problem, the equation \(2^{7x-8} = 2^6\) allowed setting exponents equal: \(7x-8 = 6\).
• This directly leads to solving for \(x\) through basic algebra.
• Breaking down the equation like this not only simplifies it but also reduces common algebraic errors.