Question

Find the indicated real \(n\)th \(\operatorname{root}(\mathrm{s})\) of a. \(n=5, a=-32\)

Short answer

-2

Step-by-step solution

Understand the concept of nth root

The nth root of a number can be calculated by raising that number to the power of 1/n. In our problem, the number is -32 and n is 5, so we want to find a number that when raised to the power of 5 equals -32.

Compute the nth root

To find the nth root, we raise -32 to the power of 1/5. This can be done with a scientific calculator. The result should be -2, because if you raise -2 to the power of 5, you get -32.

Key concepts

Radicals and Rational Exponents

Understanding the relationship between radicals and rational exponents is essential in solving an nth root problem. In math, a radical can be written with a rational exponent, as both concepts represent the same mathematical operation. The nth root of a number is the value you multiply by itself n times to get that number.

The expression for the nth root, \( \sqrt[n]{a} \), is equivalent to raising that number to the power of \( 1/n \) or \( a^{1/n} \). This concept allows us to transform a radical into an exponent form, making it easier to work with, especially when dealing with negative bases or higher roots. With a rational exponent, we express this more compactly, thus simplifying the arithmetic or algebra involved.

Real Numbers

When dealing with nth roots, it is important to know the type of numbers you're working with. Real numbers are the set of numbers that include both rational numbers (such as fractions) and irrational numbers (numbers that cannot be expressed as a simple fraction, like the square root of 2).

For the nth root calculation of negative numbers, there's an important consideration: even-powered roots of negative numbers don't result in real numbers because you can't multiply a real number by itself an even number of times and get a negative result. However, odd-powered roots of negative numbers do result in real numbers. In the given exercise, since we have the fifth root (an odd number) of -32 (a negative real number), the result is indeed a real number.

Exponentiation

Exponentiation is a mathematical operation involving two numbers, the base (\(a\)) and the exponent (\(n\)). It corresponds to repeated multiplication of the base, \(a\), exactly \(n\) times. For instance, \(a^3\) denotes the base \(a\) multiplied by itself three times.

When dealing with nth roots, exponentiation gets slightly more complex, as we are looking for what number, when raised to a certain power, gives us the original base. This is referred to as the inverse operation of exponentiation. While exponentiation quickly scales numbers up, nth root calculations scale them down to their root values. In practice, to calculate the nth root of a number, you raise the number to an exponent that is a fraction, which grants us a powerful tool to solve a variety of mathematical problems.