Problem 59 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 57-80: Simplify the expression, and rationalize the denominator when appropriate. $$ \sqrt[5]{-64} $$

Step-by-Step Solution

Answer

The simplified expression is -2.

Step 1: Identify the Root Operation

The expression given is \( \sqrt[5]{-64} \), which involves a fifth root operation. We need to determine the number that, when raised to the fifth power, equals \( -64 \).

Step 2: Determine the Base Number

To simplify \( \sqrt[5]{-64} \), we should recognize that \(-64\) is a perfect fifth power. \( -64 = (-2)^5 \) since \((-2) \times (-2) \times (-2) \times (-2) \times (-2) = -32 \times -2 = -64\).

Step 3: Apply the Fifth Root

Taking the fifth root of both sides of \((-2)^5\) gives us the original base number because raising a number to a power and taking a root are inverse operations. Thus, \( \sqrt[5]{-64} = -2 \).

Key Concepts

Fifth Root

When dealing with radical expressions that involve a fifth root, like \( \sqrt[5]{x} \), it's essential to understand the process of extracting a base number whose fifth power will give the original expression under the root symbol. In essence, finding the fifth root involves identifying the number which, when multiplied by itself five times, results in the given number. A handy association here is similar to finding square roots but involves multiplying five instances.

• To simplify \( \sqrt[5]{x} \), try to express \(x\) as a base number raised to the power of five.
• Once you have it in the form \( a^5 \), the fifth root of the whole expression will simplify directly to \( a \). This is due to the property that powers and roots cancel each other out.
• For instance, if you have \( \sqrt[5]{32} \) and know that \( 32 = 2^5 \), then the expression simplifies to \( 2 \).

Recognizing the number that is a perfect power can greatly simplify the process of taking any kind of root, particularly the fifth root.

Negative Numbers in Radicals

A major point of confusion can arise when handling radicals with negative numbers, like \( \sqrt[5]{-64} \). Fortunately, dealing with odd roots (such as a cube root or a fifth root) involves the same process regardless of the sign of the number. Negative numbers can indeed have odd roots because an odd number of negative factors will yield a negative product.

• For example, \( \sqrt[3]{-8} \) equals \( -2 \), and \( \sqrt[5]{-32} \) equals \( -2 \).
• It's vital to recognize that the negative sign can stay with the result when taking odd roots, unlike even roots where negative radicands are undefined in real number solutions.

So, when simplifying an expression like \( \sqrt[5]{-64} \), acknowledging that the results can indeed be negative helps in simplifying to \( -2 \). This knowledge can guide you through similar radical expressions.

Perfect Powers

Identifying perfect powers is a critical skill when simplifying radical expressions. Numbers like 8, 27, or 64 are considered perfect powers because they can be expressed as a smaller base raised to an exponent. Particularly in our example, knowing that \( -64 \) is really \( (-2)^5 \) makes the simplification straightforward.

• A perfect power like \( 32 = 2^5 \) implies that \( 32 \) is exactly \( 2 \) raised to the fifth power.
• Similarly, recognizing \( -64 \) as \( (-2)^5 \) allows the effective simplification of \( \sqrt[5]{-64} \).
• With this knowledge, extracting roots becomes more about pattern recognition of these base forms.

Get comfortable identifying which numbers can be rewritten as powers of others, and let this guide your simplification process. Understanding perfect powers significantly eases simplification in radical expressions.