Question

Find the indicated real \(n\)th \(\operatorname{root}(\mathrm{s})\) of a. \(n=5, a=-1\)

Short answer

The real 5th root of -1 is -1.

Step-by-step solution

Understand the question

We are asked to find the 5th root of -1. The index (n) of the root is an odd number, which means negative numbers can have real roots. Therefore, we need to find which real number raised to the power of 5 gives -1.

Apply the rules for radicals and roots

\(-1^{1/5}\) must be calculated. Anything to the power of 1 is itself, so \(-1^{1/5}\) is -1.

Write down the solution

The real 5th root of -1 is -1.

Key concepts

Understanding Radicals and Roots

Radicals and roots are fundamental concepts in mathematics that help us solve equations involving exponents.

When we talk about \(n\)th roots, we mean finding a number that, when raised to the power \(n\), results in the original number. For instance, the 5th root of a number \(a\) is a number \(b\) such that \(b^5 = a\).

Interestingly, when \(n\) is an odd number like 5, negative numbers can have real roots. This is because a negative number raised to an odd power remains negative. In our example, we want to find the 5th root of \(-1\), asking for a number that when raised to the 5th power equals \(-1\). Through the properties of odd exponents, we identify \(-1\) as this number.

Exploring Real Numbers

Real numbers include all the numbers you can think of on the number line: positive, negative, whole numbers, fractions, and irrational numbers. They do not, however, include imaginary numbers.

In our exercise, we deal with a negative real number, \(-1\). Even though negatives may seem tricky, they are straightforward to handle when finding odd roots.

• For odd roots, like the 5th root, negative real numbers produce negative results.
• This happens because odd roots maintain the sign of the original number.

Understanding this characteristic helps us easily determine that the 5th root of \(-1\) is \(-1\). This ensures our solution is a real number, maintaining consistency with the types of numbers we are working with.

Delving into Exponents

Exponents are used to express repeated multiplication. For instance, \(b^n\) means multiplying \(b\) by itself \(n\) times.

When dealing with roots, we use fractional exponents. For example, the 5th root of a number \(a\) can be written as \(a^{1/5}\). This shows the relationship between roots and exponents clearly.

Let's see how it affects solving our problem:

• The equation \(-1^{1/5}\) represents finding which number (multiplied five times) results in \(-1\).
• The answer \(-1\) is obtained because \((-1)^5 = -1\), demonstrating how exponents work to preserve the identity of the base in the case of negative numbers and odd powers.

Through this exploration, we see the power and simplicity of exponential notation in working with both whole and fractional powers.