Question

Evaluate each expression. Retain the proper number of significant digits in your answer. Roots Find each principal root without using your calculator. $$\sqrt[5]{-32}$$

Short answer

\(-2\. The principal fifth root of -32 is -2 because \( (-2)^5 = -32 \).

Step-by-step solution

Understand Principal Roots

The principal root of a number is its primary root, which can be positive or negative for odd roots. In this case, we have a fifth root, which means we are looking for a principal root that can be negative because the index is an odd number.

Factor the Radicand

Start by factoring the radicand (the number under the root). In this case, \( -32 \) can be factored into \( -2 \times -2 \times -2 \times -2 \times -2 \) because \( -2 \times -2 \times -2 \times -2 \times -2 = -32 \) and each factor of \( -2 \) is the same, satisfying the requirement for a fifth root.

Apply the Fifth Root

The fifth root of \( -2^5 \) is \( -2 \) because when \( -2 \) is raised to the fifth power, it equals the original radicand \( -32 \). Thus, the principal fifth root of \( -32 \) is \( -2 \) since \( (-2)^5 = -32 \) and it is identical with the given radicand.

Key concepts

Significant Digits

Significant digits, also known as significant figures, are crucial in the study and application of mathematics, especially when dealing with measurements and precision. They represent the number of meaningful digits in a number, starting with the first non-zero digit. When evaluating expressions, especially those involving roots, it is essential to retain the proper number of significant digits to ensure the accuracy and precision of the result.

For example, when given a number like 0.00420, there are three significant digits: 4, 2, and the final 0, because it indicates precision. In the context of performing operations with roots, if the original number has a certain number of significant digits, the result must also reflect this degree of precision. However, in some cases like integer or simple fraction radicands, significant digits don't play a critical role, as these numbers are exact by nature.

Factoring Radicands

Factoring radicands is a critical step in simplifying radical expressions and finding principal roots without the aid of a calculator. The radicand is the number inside the radical sign that we want to take the root of. In the case of a negative radicand, as seen with odd roots, the factoring process involves finding prime factors that, when multiplied together, would result in the original radicand.

For example, with the given expression \( \sqrt[5]{-32} \), we first identify \( -32 \) as our radicand. The factoring process would break down \( -32 \) into \( -2 \times -2 \times -2 \times -2 \times -2 \) since this is a repetitive multiplication of the number \( -2 \) five times. This repetitive factor directly correlates with the index of the root (which is 5 in this case), paving the way to easily identify \( -2 \) as the principal fifth root of \( -32 \). The process of factoring radicands also helps in identifying perfect powers within the radicand that match the degree of the root, simplifying the extraction of roots.

Odd Roots

Odd roots are roots with an odd index, such as the third root (cube root), fifth root, and so on. These roots are unique because they can be taken of both positive and negative numbers, unlike even roots, which can only be taken of positive numbers, or zero, to result in real numbers. When computing odd roots of negative numbers, the result will also be negative.

In the context of the exercise \( \sqrt[5]{-32} \), we are dealing with an odd root since the index of the root is 5, an odd number. It is important to recognize that odd roots follow the same rules of arithmetic as any other number. That is, a negative number raised to an odd power will result in a negative number. Thus, when you calculate the fifth root of \( -32 \) and know that \( -32 \) is the fifth power of \( -2 \) (\( (-2)^5 = -32 \) ), the principal root is immediately apparent to be \( -2 \) because it is the number that, when raised to the power of 5, gives the original radicand, \( -32 \). This understanding of odd roots is essential for correctly solving radical expressions involving odd indices.