Problem 13 - Algebra A Combined Function

Question

Use radical notation to rewrite each expression. Simplify if possible. $$ (-27)^{1 / 3} $$

Step-by-Step Solution

Answer

The simplified form is -3.

Step 1: Convert the Expression to Radical Notation

To rewrite the given expression \[ (-27)^{1/3} \]in radical notation, remember that an expression in the form of \( a^{1/n} \) can be written as \( \sqrt[n]{a} \). Hence, \[ (-27)^{1/3} \]is written as\[ \sqrt[3]{-27} \].

Step 2: Simplify the Radical Expression

The cube root of a number is a value that, when multiplied by itself three times, gives the original number. Here, we need to find the cube root of \(-27\). Since \[ (-3) imes (-3) imes (-3) = -27 \],the cube root of \(-27\) is \(-3\).

Key Concepts

Cube Root

The cube root of a number is a special type of root where we are trying to find a value that, when multiplied by itself three times, results in the original number. This is denoted by the symbol \( \sqrt[3]{a} \), where \( a \) is the number whose cube root we are finding. For example, let's think about \( \sqrt[3]{-27} \).When determining the cube root of \(-27\), we are looking for a number which fulfills \( x \times x \times x = -27 \). In this case, the number \(-3\) satisfies this condition, since \(-3 \times -3 \times -3 = -27\).Consider these additional points when working with cube roots:

• The cube root of a negative number is also negative since multiplying three negative numbers results in a negative product.
• Cube roots are not limited to integers; they may also be fractions or decimal values.

Simplifying Expressions

Simplifying expressions is the process of making expressions as concise as possible by reducing them to their simplest form. This involves performing basic operations like addition, subtraction, multiplication, and division where applicable, and combining like terms.With expressions involving radicals, such as cube roots, simplification requires identifying the factors of numbers that can be expressed as a perfect cube. For instance, if we consider \( \sqrt[3]{-27} \), it can be simplified since \(-27\) is a perfect cube (\(-3 \times -3 \times -3\)). This simplifies directly to \(-3\).To simplify expressions that aren't perfect cubes, you might end up using approximations or identifying factors that are smaller perfect cubes.

• Simplifying requires patience and practice to recognize patterns, such as perfect cubes.
• The goal is always to reach the simplest form that is easy to understand and work with further.

Rational Exponents

Rational exponents offer a different perspective for expressing roots using fractions. In mathematics, expressions in the form \( a^{m/n} \) are equivalent to \( \sqrt[n]{a^m} \), where \( n \) is the root and \( m \) is the power of \( a \).This form is incredibly flexible, allowing operations traditionally used with whole number exponents to be applied to roots. For the initial problem, \((-27)^{1/3}\), the exponent is \(1/3\). This indicates we need the cube root of \(-27\), which as we have seen simplifies to \(-3\).Remember these helpful observations about rational exponents:

• They allow for easier manipulation of expressions, combining both roots and powers in a single format.
• They make it possible to perform standard mathematical operations more seamlessly across both exponents and roots.

Start practicing by converting between radical notation and rational exponents to build a reliable skill set for simplifying diverse expressions.