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Chapter 7: Problem 94

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

Short Answer

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

Step by step solution

01

Express the Given Equation

We start with the given equation \(32^{0.8} x = 1\). The aim is to find the value of \(x\).
02

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: \[ x = \frac{1}{32^{0.8}} \]
03

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 \[ x = \frac{1}{16} \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

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.

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Most popular questions from this chapter

Rewrite each function to make it easy to graph using transformations of its parent function. Describe the graph. \(y=-2 \sqrt{49 x+49}\)

Which equation shows \(y+3=\sqrt{\frac{x}{16}+2}\) rewritten in the form \(y=a \sqrt{x-h}+k ?\) F. \(y=\frac{3}{4} \sqrt{x-(-2)}\) G. \(y=\frac{1}{4} \sqrt{x-(-2)}+(-3)\) H. \(y=\frac{1}{4} \sqrt{x-(-32)}+(-3)\) J. \(y=\frac{1}{8} \sqrt{x+32}+(-3)\)

How is the graph of \(y=\sqrt{x+7}\) translated from the graph of \(y=\sqrt{x} ?\) A. shifted 7 units left B. shifted 7 units right C. shifted 7 units up D. shifted 7 units down

Solve each square root equation by graphing. Round the answer to the nearest hundredth if necessary. If there is no solution, explain why. \(3 \sqrt{3-x}=10\)

Rewrite each function to make it easy to graph using transformations of its parent function. Describe the graph. Find the domain and the range of each function. \(y=-\sqrt{2 x+8}\)
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