Question

Explain how to use the sign of \(a\) to determine the number of real fourth roots of \(a\) and the number of real fifth roots of \(a\).

Short answer

The sign of \(a\) affects the number of real roots in the case of even roots. If \(a\) is positive or zero, there will be two real fourth roots of \(a\), but if \(a\) is negative, there will be no real fourth roots of \(a\). On the other hand, irrespective of the sign of \(a\), there will always be one real fifth root of \(a\).

Step-by-step solution

Conceptual Understanding of Roots

Firstly, it is important to comprehend that a 'root' of a number is the value that, when raised to a certain power, gives the original number. If \(n\) is an integer, then the nth root of \(a\) is a number \(x\) such that \(x^n = a\). Additionally, the term 'real root' refers to roots that are real numbers. They can be positive, negative or zero, but they cannot be imaginary or complex numbers.

Understand Effect of Even Roots

An even root of a number \(a\) is determined by raising a number to the power of 2, 4, 6, etc. Now, for a number \(a\) to have real fourth roots (or any real even roots), \(a\) must be greater than or equal to 0 (i.e., \(a\) can be positive or zero). Therefore, if \(a\) is positive or 0, there will be two real fourth roots of \(a\). If \(a\) is negative, there will be no real fourth roots of \(a\). This is because if you square any real number (positive or negative), the outcome is always a nonnegative number.

Understand Effect of Odd Roots

An odd root of a number \(a\) is determined by raising a number to the power of 3, 5, 7, etc. Unlike even roots, \(a\) can have real fifth roots (or any real odd roots) irrespective of whether \(a\) is positive, negative or zero. Therefore, irrespective of the sign of \(a\), there will always be one real fifth root of \(a\). This is because raising any real number to an odd power always results in a real number that has the same sign as the original number.

Key concepts

nth root

Understanding the concept of an nth root is essential for solving many mathematical problems. Consider a number, say a, and think about what it means to find its nth root. Formally, the nth root of a is a number which, when raised to the power of n, gives a. This can be represented by the equation: \( x^n = a \).

For instance, if we're looking for the cube root (which is also the third root), of 8, we're looking for a number that, when cubed (raised to the power of 3), equals 8. Here, the cube root is 2, as \( 2^3 = 8 \). Similarly, the fourth root of 16 is 2, as \( 2^4 = 16 \).

When dealing with nth roots, especially with larger values of n, a calculator or computer algebra system can be extremely helpful in finding precise values. However, understanding the properties and behaviors of these roots is crucial for analytical purposes.

even and odd roots

When it comes to roots, it's crucial to differentiate between even and odd roots, as they behave quite differently. For even roots, specifically when n is an even number such as 2, 4, 6, and so on, there are certain restrictions on the original number a.

Even Roots

If a is negative and we are seeking an even root, there will be no real roots, as you cannot raise a real number to an even power and obtain a negative result. For example, \( x^2 = -1 \) has no real solutions since any real number squared results in a positive number or zero.

On the other hand, if a is non-negative (positive or zero), there are exactly two real even roots (except in the case where a is 0, which has only one real even root, which is 0 itself). For instance, the equation \( x^4 = 16 \) has two real solutions: x = 2 and x = -2.

Odd Roots

Odd roots, however, are more straightforward. No matter what the sign of a is, there will always be exactly one real odd root. This is because a real number a can be raised to an odd power to produce both positive and negative results. For example, both x = -2 and x = 2 are solutions to the equation \( x^3 = -8 \), and \( (-2)^3 = -8 \) and \( 2^3 = 8 \) respectively, illustrating that negative bases raised to odd exponents result in negative results.

real numbers

Real numbers include all the numbers on the number line. This means every positive number, every negative number, zero, and every fraction or decimal number falls into the category of real numbers. They do not include imaginary or complex numbers. In the context of roots, the term 'real root' refers to roots that are, indeed, real numbers.

Understanding the set of real numbers is vital when considering the roots of a number because it dictates possible solutions. As noted in the exploration of even and odd roots, the sign of the original number a greatly influences the existence and number of real roots. For every positive real number a, there will be real nth roots, and for negative numbers a, real nth roots only exist when the root is odd. Zero, while special, is also a real number, and raising zero to any power results in zero.

Learning how to find and interpret real roots is a fundamental mathematical skill. It has applications in solving equations, analyzing graphs, and understanding the behavior of functions. Mastery of these concepts deepens one's mathematical knowledge and expands the tools available for tackling a wide array of problems.