Problem 85 - College Algebra Essentials

Question

In Exercises \(83-90\), evaluate each expression without using a calculator. $$8^{\frac{1}{3}}$$

Step-by-Step Solution

Answer

The value of \(8^{\frac{1}{3}}\) equals 2.

Step 1: Understanding Fractional Exponents

The expression \(8^{\frac{1}{3}}\) is equivalent to the cube root of 8. A fractional exponent \(\frac{1}{n}\) is equal to the nth root of a number.

Step 2: Simplify the Expression

The cube root of 8 is 2 since \(2^3 = 8\). Thus, \(8^{\frac{1}{3}} = 2\)

Key Concepts

cube root

A cube root is a special type of root in mathematics. When you take the cube root of a number, you are looking for a value that, when multiplied by itself three times, will give you the original number. It is denoted by the symbol \(\sqrt[3]{x}\). For example, the cube root of 8 is written as \(\sqrt[3]{8}\).

Here are some additional points to help you understand cube roots more clearly:

• Finding a cube root is the inverse operation of cubing a number (raising it to the third power).
• If \(a^3 = b\), then \(\sqrt[3]{b} = a\).
• Cube roots can result in both positive and negative values since both positive and negative numbers can be cubed to get the same positive or negative result.
• The simplest way to think about cube roots is to ask: "Which number times itself once and once more equals this number?"

Knowing how to efficiently find and use cube roots can make solving complex mathematical problems more manageable.

simplifying expressions

Simplifying expressions is a foundational skill in mathematics that involves reducing expressions to their simplest form. When simplification is done correctly, it can make calculations easier and make expressions easier to understand and compare.

The simplification process often involves several techniques:

• Combining like terms: Like terms have the same variables raised to the same power. These can be added or subtracted together.
• Applying the laws of exponents: Understanding how to manipulate exponents is crucial, especially when dealing with fractional exponents.
• Using factorization: Breaking down numbers or expressions into their constituent factors can reveal patterns or simpler expressions.
• Understanding mathematical properties such as the distributive property or the commutative property to reorganize and reduce expressions.

When applied to the expression \(8^{\frac{1}{3}}\), simplifying means recognizing this as the cube root of 8 and further evaluating to find it equals 2. Simplifying expressions is an invaluable tool that streamlines problem-solving and helps clarify complex algebraic structures.

nth root

The nth root of a number is a concept that extends the idea of square roots and cube roots to any integer power \(n\). When solving expressions with fractional exponents, the nth root plays a central role.

Some points to help with understanding nth roots:

• The nth root of a number \(x\), represented as \(\sqrt[n]{x}\) or as \(x^{\frac{1}{n}}\), asks the question: “What number can I multiply by itself \(n\) times to get \(x\)?”
• If \(y^n = x\), then \(\sqrt[n]{x} = y\).
• When \(n\) is an even number, there can be both positive and negative nth roots. However, the principal root is usually the positive one.
• It is often used in problems involving fractional exponents where the denominator of the exponent indicates the root.

Understanding nth roots is particularly useful in algebra and calculus, where simplifying and transforming expressions can often require manipulating roots and powers.