Problem 59 - Precalculus: Functions and Graphs

Question

Simplify the expression, and rationalize the denominator when appropriate. $$\sqrt[5]{-64}$$

Step-by-Step Solution

Answer

The simplified expression is \(-2\).

Step 1: Understand the Problem

We are given the expression \( \sqrt[5]{-64} \) and need to simplify it. This means we need to find a number that, when raised to the fifth power, equals \(-64\).

Step 2: Find the Base Number

To simplify \( \sqrt[5]{-64} \), we need to determine a number that multiplied by itself five times results in \(-64\). Recall the property of exponents that states for any real number \(x\) it holds that \(x^5 = -64\).

Step 3: Determine the Fifth Root

Consider the base number \(-2\), because \((-2)^5 = -2 \times -2 \times -2 \times -2 \times -2 = -32 \times -2 = -64\). We verify that \(-2\) raised to the fifth power indeed equals \(-64\).

Step 4: Write the Simplified Expression

The expression \( \sqrt[5]{-64} = -2 \). Since this calculation holds, the simplified expression is simply \(-2\).

Key Concepts

Negative Numbers

Negative numbers are numbers that are less than zero. These numbers have a '-' sign in front, which signifies their negative nature. When you encounter a negative number, it's important to understand how it interacts with other numbers in operations.

• Adding a negative number is the same as subtracting the number's positive equivalent.
• Subtracting a negative number is akin to adding the number's positive equivalent.
• Multiplying or dividing two negative numbers results in a positive number, while multiplying or dividing a negative number and a positive number results in a negative number.

Understanding how negative numbers behave is crucial when simplifying expressions, especially when dealing with exponents and roots. For instance, in the expression \(\sqrt[5]{-64}\), the negative sign is critical. This sign indicates that our result after simplification should still be negative. Remember that odd roots, like the fifth root, can result in negative numbers if the radicand (the number underneath the root) is negative.

Exponents and Powers

Exponents are a way to express repeated multiplication. An exponent tells you how many times to multiply a base number by itself. When we say \(2^3\), for example, it means we multiply 2 by itself three times: \[2 \times 2 \times 2 = 8\].

• The base is the number being multiplied (2 in our example).
• The exponent is the small number that tells you how many times to multiply the base (3 in our example).

Exponents and powers are used commonly in algebra to simplify expressions and convey very large or very small numbers more conveniently. In the expression \((-2)^5 = -32 \), the base is '-2', and the exponent is 5, indicating we are multiplying '-2' five times. It's important to be attentive to the minus sign when dealing with exponents, especially odd exponents, as it affects whether the final result is positive or negative.

Fifth Roots

Fifth roots are the opposite of raising a number to the fifth power. To find the fifth root of a number is to find another number which, when used as a base and raised to the power of five, results in the original number. The notation for the fifth root is\(\sqrt[5]{x}\). When simplifying expressions with roots, it's necessary to determine which number multiplies to give you the radicand when raised to the appropriate power. For the expression \(\sqrt[5]{-64}\),the number is easy to find when you understand the relationship with exponents. As we determined, \((-2)^5 = -64\),hence \(\sqrt[5]{-64} = -2\).Fifth roots are especially interesting because they can yield negative results without additional complications, as opposed to even roots, which do not yield real negative results.