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Chapter 3: Problem 70

If \(x^{4}=625,\) determine the value of \(x .[3.2]\)

Short Answer

Expert verified
The value of \(x .[3.2]\) when \(x^{4}=625\) is 5.32

Step by step solution

01

Solve for x

To find the value of \(x\) when \(x^{4}=625\), we will take the fourth root of both sides of the equation, since the fourth root is the inverse of raising to the power of four.
02

Calculate the fourth root of 625

The fourth root of 625 can be calculated using a calculator or known by heart as 5, since \(5^{4}\) equals 625. So \(x=5\).
03

Determine the value of x dot [3.2]

Given \(x=5\), the expression \(x .[3.2]\) represents a decimal form. It's equivalent to \(5 .[3.2]\), which simplifies to 5.32 since dot [3.2] represents the decimal part only.

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Key Concepts

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

Fourth Roots
When faced with an equation like \(x^4 = 625\), taking the fourth root is a method to find the value of \(x\).
Fourth roots work similarly to square roots, but instead of two identical factors multiplying to create a number, it involves four identical factors.
For instance:
  • If \(x^4 = 625\), then \(x\) is the number that, when multiplied by itself four times, results in \(625\).
  • This leads us to \(x = \sqrt[4]{625}\).
By calculating, you'll find that \(5^4 = 625\), which means the fourth root of \(625\) is \(5\). Recognizing these powers can help you solve similar problems more quickly. It's beneficial to remember the fourth roots of common numbers, much like you would with square roots.
Exponents
Exponents indicate how many times a number, known as the base, is multiplied by itself. In our given equation, \(x^4 = 625\), the exponent is \(4\).
This means \(x\) is multiplied by itself four times.
Some principles of exponents include:
  • \(x^1 = x\) - A base raised to the power of 1 is itself.
  • \(x^0 = 1\) - Any base raised to the power of 0 equals 1, given that the base is not zero.
  • \(x^n \cdot x^m = x^{n+m}\) - When multiplying like bases, you add the exponents.
  • \((x^n)^m = x^{n\cdot m}\) - When raising a power to another power, you multiply the exponents.
Exponents simplify complex multiplication, but be wary of misapplying these rules. Each rule has specific situations where it applies, especially in higher-level math problems.
Solving Equations
Solving equations often involves isolating the variable to find its value. Let's break down this process using the example \(x^4 = 625\).
You'll want to reverse the operations affecting \(x\) determining its value:
  • Start by identifying the operation affecting the variable, which is the exponent \(4\).
  • The opposite operation of raising to the fourth power is taking the fourth root. This step neutralizes the exponent effect, allowing you to solve for \(x\).
  • Calculate \(\sqrt[4]{625}\) to find \(x = 5\).
By isolating \(x\), we now know its value. A good strategy is to check your work by substituting \(5\) back into the original equation, confirming that it satisfies \(5^4 = 625\), which reassures that your solution is correct. Understanding these principles aids considerably in tackling various algebraic challenges.

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Most popular questions from this chapter

Explain how to use the graph of the first function \(f\) to produce the graph of the second function \(F\). $$f(x)=e^{x}, F(x)=e^{-x}+2$$

Use a graphing utility to graph each function. If the function has a horizontal asymptote, state the equation of the horizontal asymptote. $$f(x)=0.5 e^{-x}$$

Determine the domain of the given function. Write the domain using interval notation. $$f(x)=\frac{e^{x}-e^{-x}}{e^{x}+e^{-x}}$$

A model for how long our aluminum resources will last is given by $$T=\frac{\ln (20,500 r+1)}{\ln (r+1)}$$ where \(r\) is the percent increase in consumption from current levels of use and \(T\) is the time (in years) before the resource is depleted. a. Use a graphing utility to graph this equation. b. If our consumption of aluminum increases by \(5 \%\) per year, in how many years (to the nearest year) will we deplete our aluminum resources? c. What percent increase in consumption of aluminum will deplete the resource in 100 years? Round to the nearest tenth of a percent.

Determine the domain of the given function. Write the domain using interval notation. $$f(x)=\sqrt{1-e^{x}}$$
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