Problem 502 - Elementary Algebra

Question

In the following exercises, simplify. \(\sqrt[3]{a^{3}}\)

Step-by-Step Solution

Answer

The simplified expression is \(a\).

Step 1: Identify the Expression

The given expression is \(\sqrt[3]{a^{3}}\). It represents the cube root of \(a^3\).

Step 2: Understand the Cube Root

The cube root of a number is the value that, when multiplied by itself three times, gives the original number. Thus, \(\sqrt[3]{x^3} = x\) for any value of \(x\).

Step 3: Simplify the Expression

Given \(\sqrt[3]{a^{3}}\), apply the cube root property: \(\sqrt[3]{a^{3}} = a\). This simplifies the expression to just \(a\).

Key Concepts

Cube Root

The cube root is a fundamental concept in mathematics. It involves finding a number that, when multiplied by itself three times, results in the original number. For example, the cube root of 27 is 3 because 3 * 3 * 3 = 27.

When dealing with variables, it works the same way. In our exercise, we have \( \sqrt[3]{a^{3}} \). The cube root of \( a^{3} \) is simply \( a \) because \( a * a * a = a^{3} \). This property helps simplify complex expressions easily.

Exponents

Exponents indicate how many times a number (the base) is multiplied by itself. For example, in \( a^{3} \, a \) is the base, and \( 3 \) is the exponent, meaning we multiply \( a \) by itself three times: \( a * a * a \).

Understanding exponents makes simplifying expressions like \( \sqrt[3]{a^{3}} \) much more manageable. Here, recognizing that 3 in \( a^{3} \) matches the 3 of the cube root, we easily deduce that the result is \( a \).

Exponents follow specific rules that can simplify many complex algebraic expressions. One key rule used here is that the cube root (\[ \sqrt[3]{x^{n}} = x^{n/3} \]) cancels out the exponent if they are equal.

Simplification

Simplifying radical expressions is an essential skill. It involves breaking down expressions into their simplest forms using mathematical properties and rules.

The goal is to make complex expressions more manageable and easier to understand. For the given problem, \( \sqrt[3]{a^{3}} \), we identified that the cube root and the exponent are the same, allowing for a direct simplification: \( \sqrt[3]{a^{3}} = a \).

In general, simplification can follow various approaches based on the given expression. Understanding properties of roots and exponents is critical in applying these techniques correctly.

By mastering simplification, you form a foundation for tackling more advanced problems in algebra and calculus.