Problem 87 - College Algebra Essentials

Question

In Exercises \(83-90\), evaluate each expression without using a calculator. $$125^{\frac{2}{3}}$$

Step-by-Step Solution

Answer

The value of \(125^{\frac{2}{3}}\) is 25.

Step 1: Re-write the expression

The given expression \(125^{\frac{2}{3}}\) can be rewritten using the rule that \(a^{\frac{m}{n}} = (\sqrt[n]{a})^{m}\), where a is any real number and m and n are positive integers. Thus, \(125^{\frac{2}{3}}\) can be written as \((\sqrt[3]{125})^{2}\).

Step 2: Take the cube root

The cube root of 125 is 5, because \(5^3 = 125\). So \((\sqrt[3]{125})^{2}\) is equal to \(5^{2}\).

Step 3: Square the result

\((\sqrt[3]{125})^{2}\), or \(5^{2}\), is equal to 25.

Key Concepts

Cube Root

The cube root of a number, represented as \( \sqrt[3]{x} \), is a value that, when multiplied by itself three times, gives the original number, x. In other words, if we have \( \sqrt[3]{a} = b \), then it must be true that \( b^3 = a \). The cube root is particularly special because it deals with three dimensions, akin to the three dimensions of a cube.

Finding the cube root without a calculator involves recognizing numbers that are perfect cubes or using methods like prime factorization. For example, the cube root of 125 is 5 because \( 5 \times 5 \times 5 = 125 \) or \( 5^3 = 125 \). Understanding cube roots is fundamental in evaluating expressions with fractional exponents and is commonly used in geometry, physics, and engineering calculations.

Exponents

Exponents represent repeated multiplication. When you see \( a^n \), it means you multiply the base, a, by itself n times. If n is a positive integer, you get straightforward multiplication, like \( a^3 = a \times a \times a \).

However, exponents can also be fractions, which indicates roots. For example, \( a^{\frac{1}{n}} \) means the n-th root of a and \( a^{\frac{m}{n}} \) means the n-th root of a, raised to the m-th power. An exponent can even be negative, which signifies a reciprocal, like \( a^{-n} = \frac{1}{a^n} \). Understanding how to manipulate exponents is crucial for simplifying expressions and solving algebraic equations.

Radicals

Radicals involve taking roots of numbers, such as square roots, cube roots, and higher. The radical symbol \( \sqrt[n]{...} \) specifies the type of root. The n inside the bracket is the root's degree, and the value inside is the radicand. For most students, the square root (where n=2), is the most familiar.

A key to working with radicals is recognizing when a radicand is a power of the root degree, which simplifies the expression entirely out of the radical. For example, \( \sqrt[3]{125} \) simplifies to 5, since 125 is a perfect cube. When dealing with radicals, important principles to remember are the multiplication and division of roots, as well as rationalizing the denominator to write expressions in their simplest form.

Simplifying Expressions

Simplifying expressions is the process of altering a complex algebraic expression into a simpler or more readable form without changing its value. The steps often include combining like terms, using the distributive property, and simplifying radicals or exponents.

For example, in solving \(125^{\frac{2}{3}}\), we simplified the complex exponential expression first by translating it into a radical expression and then by taking the cube root and squaring the result. By recognizing that certain exponents represent roots, and by understanding how to work with roots, you can greatly simplify what may initially appear to be a daunting expression. Mastery of simplification techniques is essential, as it allows for easier evaluation and can make the difference in successfully solving more complex algebraic problems.