Problem 51 - Algebra 2

Question

Simplify each number. $$(-243)^{\frac{1}{5}}$$

Step-by-Step Solution

Answer

The simplified form of \((-243)^{\frac{1}{5}}\) is \(-3\).

Step 1: Understand the power

The provided number is \(-243\) and the power is \(\frac{1}{5}\). This can be understood as taking the 5th root of \(-243\).

Step 2: Calculate the 5th root

As the 5 is an odd number, taking the fifth root of a negative number is possible. The fifth root of \(-243\) is \(-3\) as when \(-3\) is multiplied by itself five times, it results in \(-243\).

Step 3: Final answer

Therefore, the simplified form of \((-243)^{\frac{1}{5}}\) is \(-3\).

Key Concepts

Understanding the Fifth Root

The fifth root of a number is a special type of root, similar to the square or cube root. In general, the nth root of a number is what you multiply by itself n times to get that number back. For the fifth root, imagine asking: "What number do I multiply by itself five times to get the original number?" When dealing with negative numbers and odd roots, such as the fifth root, it's essential to realize you can indeed have a negative result. This is because multiplying an odd number of negative factors will yield a negative product. Hence, when calculating the fifth root of -243, we find that -3 is the correct answer because \((-3) \times (-3) \times (-3) \times (-3) \times (-3) = -243\).

Working with Negative Numbers

Negative numbers are numbers less than zero, and they play a significant role in mathematics, especially when working with roots and exponents. It's sometimes tricky to compute roots of negative numbers, but with odd roots such as the fifth root, it is straightforward. Negative numbers, when multiplied by themselves an odd number of times, will result in a negative product. However, the typical rules of arithmetic for negative numbers still apply. When multiplying or dividing two negative numbers, the result will be positive, while the product of a positive and a negative number will always be negative. Remember, simplifying expressions with negative numbers often involves careful attention to signs.

Exploring Rational Exponents

Rational exponents are another way to express roots. Instead of writing the fifth root of -243, for example, you can write \((-243)^{\frac{1}{5}}\). This showcases a connection between powers and roots. In general, \(x^\frac{1}{n}\) means to take the nth root of x. These exponents are called rational because they can be written as fractions. The numerator of this fraction represents the power, and the denominator represents the root. Therefore, a rational exponent of \(\frac{1}{5}\) signifies a fifth root. Understanding this notation makes solving and simplifying expressions with roots much more convenient, as you'll often see them expressed in this form in algebra.

Basic Algebraic Operations

Algebraic operations form the backbone of simplifying expressions. They include addition, subtraction, multiplication, and division—all governed by specific rules, especially when involving exponents and roots. When dealing with expressions like \((-243)^{\frac{1}{5}}\), focus first on the exponent. Understand its role, then perform the root operation. Additionally, operations with radicals and exponents require following the order of operations—beginning with brackets or parentheses, then exponents, followed by multiplication or division, and lastly, any addition or subtraction. Carefully applying these operations allows for the successful simplification of algebraic expressions involving roots and rational exponents.