/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 18 \(x^{4}-2=0\)... [FREE SOLUTION] | 91影视

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Chapter 4: Problem 18

\(x^{4}-2=0\)

Short Answer

Expert verified
The solution to the equation \(x^{4} -2 =0\) is x = 卤1.189.

Step by step solution

01

Arrange Equation

First, we need to arrange the equation to isolate \(x^{4}\) on one side of the equation. This gives us \(x^{4} = 2\)
02

Take Fourth Root of Both Sides

To solve for x, take the fourth root of both sides: \(x = \sqrt[4]{2}\). Since the number 4 is an even number we make sure to note that there is a positive and negative solution. So, the solutions are: x= \(\sqrt[4]{2}\) and -\(\sqrt[4]{2}\).
03

Evaluate

Use a calculator, if needed, to get the approximate decimal value: x = 卤1.189.

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Key Concepts

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

Polynomial Equations
A polynomial equation is an expression consisting of variables and coefficients, which involves only the operations of addition, subtraction, multiplication, and non-negative integer exponents of variables. In simple terms, it is an equation that has more than one "term". These equations can vary in degree, which means the highest power of the variable. For instance, a polynomial of degree 4 means the highest exponent of x is 4. In our exercise, the equation \( x^4 - 2 = 0 \) is a polynomial equation of degree 4.
  • The variable in the equation is \( x \).
  • The highest power of \( x \) is 4, making it a fourth-degree polynomial.
  • The equation includes subtraction between a polynomial term (\( x^4 \)) and a constant (2).
Understanding the structure of polynomial equations is crucial. It helps us recognize patterns and methods for solving, such as factoring or finding roots.
Roots of Equations
The "roots" or "solutions" of an equation are the values of the variable that satisfy the equation. In other words, when substituted back into the equation, they make it true. For the polynomial equation \( x^4 - 2 = 0 \), the roots are the values of \( x \) that make the equation equal zero.
  • In this case, the equation \( x^4 = 2 \) is a pivotal step, as it sets the stage for finding the roots.
  • The roots are obtained by taking the fourth root of both sides of the equation.
  • Since the exponent is even, there are both positive and negative roots: \( x = \sqrt[4]{2} \) and \( x = -\sqrt[4]{2} \).
Roots are integral in determining the behavior of polynomial graphs. Knowing whether they are real, repeated, or complex impacts how the graph of the polynomial looks.
Solving Equations
Solving equations involves finding values of the variables that satisfy the equation. In algebra, especially with polynomial equations, the process can include various steps such as simplification, factoring, or using arithmetic operations. In the exercise given, we employ a property of exponents to find the solutions.Start by rearranging the equation to simplify your work:
  • From \( x^4 - 2 = 0 \), rearrange to \( x^4 = 2 \).
  • Next, solve for \( x \) by applying the inverse procedure of raising to the power, which is finding the fourth root: \( x = \sqrt[4]{2} \).
When dealing with even powers, remember that solutions are usually accompanied by a positive and a negative thanks to the symmetry of even functions on the graph. Solving such equations builds confidence in algebraic manipulations and demonstrates the importance of inverse operations in mathematics.

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Most popular questions from this chapter

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