Problem 63 - College Algebra Essentials

Question

Evaluate each expression in Exercises \(55-66,\) or indicate that the root is not a real number. $$\sqrt[5]{(-3)^{5}}$$

Step-by-Step Solution

Answer

-3

Step 1: Write the expression

The given expression is \(\sqrt[5]{(-3)^{5}}\).

Step 2: Simplify the expression

Looking at the expression, it is evident that a number raised to a power, and then taking the root of that power is a reverse operation. The fifth root of a number raised to the power of 5 is the original number. In this case, the number is -3. So, \(\sqrt[5]{(-3)^{5}} = -3\).

Key Concepts

Real Numbers

Real numbers are a central part of algebra and everyday mathematics. They combine both rational and irrational numbers. Rational numbers are fractions or integers like \(-2, 0,\) and \(0.75\). Irrational numbers include numbers that cannot be expressed as simple fractions, such as \(\sqrt{2}\) or \(\pi\). Together, these numbers form the continuum of real numbers that you can visualize on a number line.

• Real numbers can be positive, negative, or zero.
• They include special subsets like whole numbers, integers, and decimal numbers.
• The concept of real numbers helps us solve equations and model real-world scenarios effectively.

When dealing with expressions like \[\sqrt[5]{(-3)^{5}}\], understanding that\(( -3 )\) is a real number assists in simplifying the expression, as real numbers are stable under operations like roots and exponents.

Exponents

Exponents are a shorthand way of expressing repeated multiplication of the same number by itself. For example, \(3^5\) means multiplying 3 by itself five times: \(3 \times 3 \times 3 \times 3 \times 3\).

• An exponent tells you how many times the base is multiplied.
• Negative exponents represent reciprocal values, like \(3^{-2} = \frac{1}{3^2}\).
• Zero exponents also have a special rule: any nonzero number raised to the power zero is 1, written as \(n^0 = 1\).

In the expression \[\sqrt[5]{(-3)^{5}}\], the exponent is 5. The process essentially reverses itself when you then apply the fifth root, explaining why the simplified result is \(-3\). This operation demonstrates how exponents and roots can be inverse operations.

Roots

Roots, particularly square roots and higher roots, are the opposite operation to raising a number to a power. The notation \(\sqrt[n]{x}\) signifies the \(n\)th root of \(x\), and it is essentially asking, "What number raised to the \(n\)th power gives \(x\)?"

• Finding a square root is common for \(n = 2\), denoted simply as \(\sqrt{x}\).
• Higher roots, such as cube roots or fifth roots, are used similarly.
• In the real numbers, only even roots of negative numbers are not real, whereas odd roots can be negative.

In the specific case of \[\sqrt[5]{(-3)^{5}}\], the fifth root cancels out raising \(-3\) to the fifth power, indicating that the number under the radical is indeed real and equal to \(-3\). This showcases how roots can simplify expressions when combined with their respective powers.