Problem 94 - Algebra 2

Question

What is the value of \(x\) if \(32^{0.8} x=1 ?\) Simplify the answer.

Step-by-Step Solution

Answer

The value of \(x\) in the given equation is \(\frac{1}{16}\).

Step 1: Express the Given Equation

We start with the given equation \(32^{0.8} x = 1\). The aim is to find the value of \(x\).

Step 2: Isolate the 'x'

In order to solve for \(x\), we should isolate it. We can do this by dividing both sides of the equation by \(32^{0.8}\). Like this: \[ x = \frac{1}{32^{0.8}} \]

Step 3: Simplify the Value

The expression can be simplified by solving the exponent. In this case, \(32^{0.8}\) can be rewritten as \((2^5)^{0.8} = 2^{5*0.8} = 2^4 = 16\), giving us \[ x = \frac{1}{16} \]

Key Concepts

Exponent Properties

Understanding exponent properties is crucial when solving exponential equations like the one in this exercise. Exponents indicate how many times a number, known as the base, is multiplied by itself. A simple example is that in the term \( a^b \), \( a \) is the base and \( b \) is the exponent. Here, the base \( a \) is raised to the power of \( b \).
When dealing with exponents, certain rules make calculations easier:

• Product of Powers: For the same base, you can add exponents: \( a^m \times a^n = a^{m+n} \).
• Power of a Power: Multiply exponents: \( (a^m)^n = a^{m\cdot n} \).
• Quotient of Powers: Subtract exponents: \( \frac{a^m}{a^n} = a^{m-n} \).
• Zero Exponent: Any base raised to zero equals one: \( a^0 = 1 \).
• Negative Exponent: Turn into a reciprocal: \( a^{-m} = \frac{1}{a^m} \).

In our exercise, the expression \(32^{0.8}\) uses the power of a power rule, where the base 32 is rewritten as \((2^5)^{0.8}\). By multiplying exponents, it further simplifies to \(2^4\), which equals 16.

Isolating Variables

Isolating the variable is a key step in solving equations. The goal is to have the variable on one side of the equation, all by itself. This process makes it possible to determine its value.
In the equation \(32^{0.8} x = 1\), our aim is to solve for \(x\). The simplest way to isolate \(x\) is to perform the opposite operation of what is currently being applied to it. Here, \(x\) is multiplied by \(32^{0.8}\), meaning we should divide both sides by \(32^{0.8}\).
This step ensures:

• The variable \(x\) stands alone on one side: \(x = \frac{1}{32^{0.8}}\).
• The other side of the equation becomes the value representing \(x\).

By performing this division, we are simplifying the problem, leading us to the next step of evaluating or simplifying the expression accordingly.

Simplifying Expressions

Simplifying expressions is the process of making them as straightforward as possible. After isolating \(x\) in our original equation, the result was \(x = \frac{1}{32^{0.8}}\).
To simplify, it's helpful to first express the base in a simpler form. Here, 32 can be rewritten as \(2^5\), which is a smaller base with a known power. Using exponent properties, we further simplify:

• Convert \(32^{0.8}\) into \((2^5)^{0.8}\).
• By applying the power of a power rule, multiply the exponents: \(2^{5 \cdot 0.8}\), resulting in \(2^4\).
• Since \(2^4\) equals 16, we now have \(x = \frac{1}{16}\).

Thus, \(x\) simplifies neatly to \(\frac{1}{16}\).
Breaking complex expressions into smaller parts allows us to see the solution clearly. It makes calculations more manageable and ensures accurate results. Always aim to simplify step by step to avoid mistakes.