Problem 51 - Precalculus : Building Concepts and Connections

Question

How many solutions of the equation \(u^{n}=z\) are real numbers if \(n\) is odd and \(z\) is real (that is, the imaginary part of \(z\) is zero)?

Step-by-Step Solution

Answer

So, the given equation \(u^n = z\) where \(n\) is odd and \(z\) is a real number will have exactly one real solution.

Step 1: Understanding the equation

The given equation is \(u^n = z\), where \(n\) is an odd number and \(z\) is a real number. This equation is an exponent equation, wherein the variable \(u\) is raised to the odd power \(n\).\n\nSince \(n\) is an odd number, it implies that there will be odd powers of \(u\) equal to the real number \(z\). Furthermore, since \(z\) is a real number, the solutions to this equation will also be real numbers.

Step 2: Understanding odd powers

Now, the distinctive term here is that \(n\) is odd. Odd powers of a variable have a specific property which is that the sign of the original number is preserved.\n\nThis means that if \(u > 0\), then \(u^n > 0\), and if \(u < 0\), then \(u^n < 0\). This property is unique to odd powers, unlike even powers which result in positive values regardless of the sign of the original number.

Step 3: Finding the number of real solutions

Due to the properties of real numbers and uniqueness of odd powers, the equation \(u^n = z\) will have exactly one real solution for each real \(z\).

Key Concepts

Odd Powers

When we talk about odd powers, we are referring to situations where a number is raised to a power that is an odd integer, such as 1, 3, 5, etc. An important characteristic of odd powers is that they preserve the sign of the base number. This means:

• If a number is positive and you raise it to an odd power, the result remains positive.
• If a number is negative, the result will still be negative when raised to an odd power.
• If the number is zero, raising it to any power, including an odd power, results in zero.

This might differ from even powers, like squaring a number, where the negative becomes positive. This property of odd powers ensures that when solving the equation \( u^n = z \), where \( n \) is odd, there is a consistent relationship maintained between the sign of \( u \) and \( z \). Thus, if \( z \) is positive, \( u \) must also be positive, and if \( z \) is negative, \( u \) must be negative. This clear distinction leads to unique solutions.

Exponent Equations

Exponent equations are equations where variables appear as exponents or powers. This is a significant part of algebra and helps in many scientific calculations. In our context, the equation we deal with is \( u^n = z \). Here, \( u \) raised to the power of \( n \) equals a real number \( z \). To solve such equations, understanding the nature of the powers involved is crucial.

• First, analyze the power, \( n \). Knowing whether \( n \) is odd or even greatly influences the number and type of solutions.
• For odd powers, the equation often simplifies as the sign and magnitude are straightforward and mirror the input behavior.

In our case, with such equations, finding \( u \) involves taking the n-th root of \( z \). Since odd roots of a number always give a real result, the solution can be uniquely identified for any real \( z \). Familiarity with manipulating exponents allows students to handle more complex algebraic problems.

Real Numbers

Real numbers are all the numbers on the number line, encompassing both rational and irrational numbers, positive and negative numbers, including zero. They form the basis of most calculations in elementary algebra. Understanding real numbers helps in appreciating the solutions of equations like \( u^n = z \), where \( z \) needs to be a real number.

• Any number without an imaginary component, such as \( \sqrt{2} \) or \( \pi \), is real.
• For an equation to have a real solution, both its parameters and resultant value should be rooted in real numbers.

Given a situation where an equation has real solutions, it often means the results will make sense within our typical real-world contexts. For our specific equation, this implies that regardless of what \( z \) is, provided it is real, \( u \) as the solution will also be real. This underline the critical nature of understanding real numbers in broader mathematical exercises and problems.