Solve For X In The Equation X 4 7

Solving for x in the Equation x⁴ = 7: A Complete Guide

At first glance, the equation x⁴ = 7 appears beautifully simple. It asks a direct question: what number, when raised to the fourth power, equals seven? Yet, this straightforward query opens a door to profound mathematical concepts, revealing that the answer is not a single number but a set of four distinct solutions. This guide will walk you through the complete process of solving for x, transforming a basic algebraic problem into a comprehensive lesson on roots, real numbers, and the essential nature of complex numbers. You will learn not only how to find these solutions but also why there are exactly four, building a deeper intuition for polynomial equations.

Understanding the Equation: What Does x⁴ = 7 Mean?

The equation x⁴ = 7 is a polynomial equation of degree four. In its most basic interpretation within the realm of real numbers (the numbers on the standard number line), we are looking for the fourth root of 7. The fourth root of a number a is a number b such that b⁴ = a. For positive numbers like 7, there is one positive real fourth root. However, because raising a negative number to an even power also yields a positive result (e.g., (-2)⁴ = 16), there is also a corresponding negative real fourth root. This immediately tells us we have at least two real solutions.

To express these solutions formally, we use radical notation. The principal (positive) fourth root of 7 is written as ⁴√7. Therefore, the two real solutions are:

• x = ⁴√7
• x = -⁴√7

But is that all? The journey to the full solution set requires us to expand our number system slightly.

Step-by-Step Solution: Finding All Four Roots

Step 1: Isolate the Power and Take the Fourth Root

The equation is already in the form where the variable term is isolated. To undo the fourth power, we apply the fourth root to both sides of the equation. Remember, taking an even root introduces a ± symbol to account for both the positive and negative roots. x⁴ = 7 ⁴√(x⁴) = ⁴√7 |x| = ⁴√7

This absolute value statement confirms our two real solutions: x = ±⁴√7. Numerically, ⁴√7 ≈ 1.626. So, x ≈ 1.626 and x ≈ -1.626 are valid solutions you can check on a calculator.

Step 2: Introducing Complex Numbers for the Complete Picture

The Fundamental Theorem of Algebra states that a polynomial equation of degree n has exactly n complex roots (counting multiplicity). Our degree-4 equation must, therefore, have four roots in total. We have found two real roots. The other two must be complex numbers—numbers that include the imaginary unit i, defined as i = √(-1).

To find these, we reframe the equation. x⁴ = 7 can be rewritten as: x⁴ - 7 = 0 or x⁴ = 7 * (cos 0° + i sin 0°) [using polar form for the real number 7].

We now solve for the fourth roots of the complex number 7∠0° (in polar form). Using De Moivre's Theorem, the n-th roots of a complex number r∠θ are given by: ⁿ√r ∠ (θ + 360°k)/n, for k = 0, 1, 2, ..., n-1.

For our case, n=4, r=7, θ=0°. The magnitude of each root is ⁴√7. The angles