Problem 17 - Algebra 2

Question

Use a table to solve each equation. Round to the nearest hundredth. $$ 2^{x+3}=512 $$

Step-by-Step Solution

Answer

\(x = 6\)

Step 1: Rewrite 512 as a power of 2

Rewrite \(512\) as \(2^9\). So, the equation becomes: \(2^{x+3}=2^9\).

Step 2: Equate the exponents

Since the bases are equal, we can equate the exponents. So, establish the equation: \(x+3=9\).

Step 3: Solve for x

Subtract \(3\) from both sides: \(x = 9 - 3 = 6\).

Step 4: Round to the nearest hundredth

As \(x\) is an integer, rounding isn't necessary in this case. The result is the same: \(x = 6\).

Key Concepts

Power of Two

Understanding the concept of a "power of two" is essential when dealing with exponential equations involving the base 2. A power of two essentially refers to numbers that can be expressed as an exponentiation of the number 2. For example, numbers like 2, 4, 8, 16, and so on are powers of two because they can be written as \(2^1, 2^2, 2^3, 2^4\), respectively.
In the exercise problem we are dealing with, we needed to express 512 as a power of two in order to simplify and solve the equation. We find that 512 can be expressed as \(2^9\).
Knowing how to express numbers as a power of two is extremely useful for solving equations that have a common base because it allows us to directly compare exponents.

Equation Solving

When it comes to solving equations, the goal is to find the value that makes the equation true. In equations involving exponents and powers of two, we often need to manipulate the equation so that both sides have the same base. If we can achieve this, it becomes much easier to solve the equation, as we can then simply compare the exponents.
In our example, after expressing 512 as a power of two, we had the equation \(2^{x+3} = 2^9\). Since the bases are the same, we equate the exponents to form a simpler linear equation: \(x + 3 = 9\). This approach of reducing the equation to an equality of exponents simplifies the problem significantly and sets up an easy path to find the solution.

Exponents

Exponents are a shorthand way of expressing repeated multiplication. For example, \(2^3\) means \(2 \times 2 \times 2 = 8\). In the context of our exercise, we dealt with exponents to simplify and solve an equation.
Exponents follow specific rules:

• Multiplying powers with the same base involves adding the exponents: \(a^m \times a^n = a^{m+n}\).
• Dividing powers with the same base involves subtracting the exponents: \(a^m \div a^n = a^{m-n}\).
• Raising a power to another exponent involves multiplying the exponents: \((a^m)^n = a^{m \times n}\).

Understanding these rules helps greatly in manipulating and simplifying equations, especially when you need to solve for unknown variables represented as exponents.

Integer Solutions

Integer solutions refer to the type of solutions for equations where the expected result is a whole number, free of fractions or decimals. Such solutions are important in many mathematical contexts, especially those involving counting or discrete values.
In this exercise, once we solve the equation \(x+3 = 9\) for \(x\), we find \(x = 6\). Since 6 is a whole number, it qualifies as an integer solution. This kind of solution is straightforward and the problem does not require rounding or dealing with fractional parts.
Integer solutions are often the simplest type to deal with, as they avoid complexities that sometimes arise when solutions are non-integers. They give clear, definitive answers needed in many practical scenarios.