Problem 39 - Algebra 2

Question

Solve each equation. Check your solution. $$ 2^{3 x+5}=128 $$

Step-by-Step Solution

Answer

The solution is \( x = \frac{2}{3} \).

Step 1: Understand the Problem

We have the exponential equation \( 2^{3x+5} = 128 \). Our goal is to solve for \( x \) and check if our solution is correct.

Step 2: Express Right Side as a Power of 2

To solve the equation, we need to express 128 as a power of 2. We know that \( 128 = 2^7 \). This helps to rewrite the equation with the same base on both sides.

Step 3: Set Exponents Equal

Since both sides of the equation are powers of 2 (\( 2^{3x+5} = 2^7 \)), we can set the exponents equal: \[ 3x + 5 = 7 \]

Step 4: Solve for x

Subtract 5 from both sides: \[ 3x = 7 - 5 \] \[ 3x = 2 \] Then, divide both sides by 3: \[ x = \frac{2}{3} \]

Step 5: Check the Solution

Substitute \( x = \frac{2}{3} \) back into the original equation to verify: \[ 2^{3\left(\frac{2}{3}\right)+5} = 2^7 \] Simplify the exponent: \[ 2^2 + 5 = 2^7 \] \[ 2^7 = 2^7 \] Both sides are equal, confirming that \( x = \frac{2}{3} \) is the correct solution.

Key Concepts

Exponent Properties

Exponent properties are fundamental rules that allow us to manipulate expressions involving powers. They help to simplify calculations and solve exponential equations like the one in our exercise. Let's go over some key exponent properties that are useful in solving equations:

• Same Base Multiplication: When multiplying powers with the same base, add the exponents. For example, \( a^m \times a^n = a^{m+n} \).
• Division of Powers: When dividing powers with the same base, subtract the exponents. For example, \( \frac{a^m}{a^n} = a^{m-n} \).
• Power to a Power: When raising a power to another power, multiply the exponents. For example, \((a^m)^n = a^{m\cdot n}\).
• Zero Exponent: Any number raised to the power of zero is 1, which means \( a^0 = 1 \), as long as \( a eq 0 \).

Using these properties can greatly simplify your work and help find solutions effectively. By understanding the rules, the concept of equating exponents becomes easier, especially when the same base is present on each side of the equation.

Equations

Equations are mathematical statements that assert the equality of two expressions. An equation becomes particularly interesting when it involves variables, such as in our initial problem where an exponential component is linked to a variable like \( x \).When dealing with exponential equations, our goal is often to isolate the variable in the exponent. This is achieved by expressing both sides of the equation with the same base, which allows us to set the exponents equal to one another. Our exercise walks us through this process in great detail.Firstly, we ensured both sides of \( 2^{3x+5} = 128 \) were expressed using the same base, \( 2 \). This allowed us to equate the exponents directly. From there, simple arithmetic was used to solve for \( x \). Checking solutions is crucial in verifying correctness, as it's possible to miscalculate along the way. By substituting \( x \) back into the original equation, we verify that our steps were correct and ensure our understanding of the solution process.

Powers of 2

Powers of 2 are quite common in exponential equations due to their simplicity and relevance in binary systems like computing. They follow a sequence where each power represents a doubling from the previous one:

• \( 2^0 = 1 \)
• \( 2^1 = 2 \)
• \( 2^2 = 4 \)
• \( 2^3 = 8 \)
• \( 2^4 = 16 \)
• \( 2^5 = 32 \)
• \( 2^6 = 64 \)
• \( 2^7 = 128 \)

In our exercise, recognizing that \( 128 \) is \( 2^7 \) allowed us to align the equation on a base of 2, simplifying the problem significantly. Using powers of 2 often aids in breaking down complex exponential equations where technology or binary logic might be involved, highlighting the practical applications of exponential equations in computing and mathematics.