Problem 13 - Intermediate Algebra

Question

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

Step-by-Step Solution

Answer

The expression simplifies to -3.

Step 1: Understand Radical Notation

The expression is given in exponential form as \[(-27)^{1/3}\] To write it in radical notation, remember that \[a^{1/n} = \sqrt[n]{a}\]This means we need to express the cube root of -27.

Step 2: Write in Radical Form

Using the rule from Step 1, \[(-27)^{1/3}\] can be rewritten using radical notation as \[\sqrt[3]{-27}\] This notation means 'the cube root of -27'.

Step 3: Simplify the Radical Expression

We need to find the cube root of -27. The cube root of a negative number is negative, and since \[(-3) \times (-3) \times (-3) = -27\] we have: \[\sqrt[3]{-27} = -3\] So the expression simplifies to -3.

Key Concepts

Cube Root

The cube root of a number is an operation that determines what number, when multiplied by itself twice (in total, three times), will equal the given number. When dealing with a cube root, a number inside the radical symbol, we are essentially asking the question: "What number times itself three times gives me this result?"

It can be denoted as \(\sqrt[3]{a}\) where 3 is the index, indicating the cube root. In our example, \(\sqrt[3]{-27}\) represents "what number cubed equals -27?".

Recalling cube roots is crucial, especially when the number is negative. A useful tip is recognizing patterns in small integer powers:

• The cube root of -27 is -3, since \((-3) \times (-3) \times (-3) = -27\).
• The cube root and cube function are inverse operations.

Exponential Form

Exponential form is a way to express numbers by raising them to a fraction or whole number power. The general idea is that a number, referred to as the base, is increased by multiplying itself a certain number of times. This is represented as \(a^n\), where \(a\) is the base and \(n\) is the exponent.

In the given problem, \((-27)^{1/3}\), \(-27\) is the base, and \(1/3\) is the exponent. This indicates we are supposed to find the cube root of \(-27\).

Some key points about exponential form include:

• An exponent of \(1/3\) corresponds to taking the cube root.
• Exponential notation provides a compact way to express roots and repeated multiplication.
• It helps in simplifying expressions by converting between radical and exponential forms.

Simplifying Expressions

Simplifying expressions means breaking down a complex expression into the simplest form. For the task at hand, we rewrite the expression to make calculations manageable while retaining the expression’s original value.

For \((-27)^{1/3}\), which equals \(\sqrt[3]{-27}\), simplifying this requires evaluating the cube root of \(-27\).

• Knowing that \(-3 \times -3 \times -3 = -27\), we determine \(\sqrt[3]{-27} = -3\).
• Simplifying often involves recognizing known powers and roots.
• Solving cube roots can include factoring numbers if straightforward calculations become complex.

With the final simplified form, expressions are easily understood and verified.

Negative Numbers

Negative numbers are numbers less than zero and are usually denoted with a minus sign. Working with negative numbers, especially when dealing with roots and powers, follows specific rules.

When computing the cube root of a negative number, such as \(-27\), it's important to recognize that the results remain in the negative form due to the properties of multiplication:

• The cube of a negative number is always negative, explaining \((-3)^3 = -27\).
• Cube roots of negative numbers will yield negative results due to the symmetry of the operation.
• Handling negative bases and fractional exponents like \((-27)^{1/3}\) is crucial for accurate results.

Understanding these rules ensures that working with negative numbers, particularly in roots and powers, remains intuitive and systematic.