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Chapter 10: Problem 13

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

Short Answer

Expert verified
(-27)^{1/3} = -3

Step by step solution

01

Understand the Radical Notation

First, convert the expression \((-27)^{1/3}\) into radical form. The exponent \(\frac{1}{3}\) indicates a cube root. So, \((-27)^{1/3}\) in radical form is \(\sqrt[3]{-27}\).
02

Solve the Cube Root

Calculate the cube root \(\sqrt[3]{-27}\). Think of a number that when multiplied by itself three times gives \(-27\). We know that \(-3 \times -3 \times -3 = -27\). Thus, \(\sqrt[3]{-27} = -3\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Cube Root
A cube root is a special type of root in mathematics. It tells us which number, when multiplied by itself three times, results in a given number. In this case, we're dealing with the expression \( (-27)^{1/3} \). This particular expression is asking us to find a number that multiplied by itself three times equals \( -27 \).

To find the cube root of a number:
  • Look for a number that can be cubed to match the value.
  • Make sure that your result is negative if the original number is negative, because a negative number cubed will also be negative.
For \( -27 \), the number \(-3\) fits the bill because \(-3 \times -3 \times -3 = -27\). Hence, the cube root of \(-27\) is \(-3\). This means that the expression using radical notation \( \sqrt[3]{-27} \) results in \(-3\).
Simplifying Expressions
Simplifying expressions involves breaking down complex mathematical expressions into their simplest form. This can include combining like terms, reducing fractions, or translating expressions into simpler radical or exponential forms.

For expressions involving roots and exponents, such as \((-27)^{1/3}\), simplification often means rewriting the expression in a form that is easier to understand or compute.

Here are steps to simplify expressions:
  • Identify if the expression can be changed into a root or rational exponent.
  • Evaluate the simplified form, ensuring all parts of the expression are reduced as much as possible.
  • Consider the signs and their impact, especially if dealing with cube roots, which can be negative.
In the given exercise, starting with \((-27)^{1/3}\) and transforming it to \(\sqrt[3]{-27}\) helps clarify the expression by showing its radical form. Then evaluating the cube root further simplifies it to \(-3\).
Exponents in Algebra
Exponents are a foundational concept in algebra. They represent repeated multiplication of a base number. In the expression \((-27)^{1/3}\), the exponent is \(1/3\), which indicates a cube root.

Understanding exponents involves:
  • Recognizing that a fractional exponent like \(1/3\) correlates to taking the cube root of the base.
  • Knowing how to switch between radical and exponential notations. For example, \(\sqrt[3]{x}\) can be rewritten as \(x^{1/3}\).
  • Applying properties of exponents, such as \(a^{m/n}\) being equivalent to \(\sqrt[n]{a^m}\), which helps in simplifying expressions.
Practicing these conversions and simplifications strengthens algebraic problem-solving skills. In algebra, being comfortable with both radical and exponential forms is essential, as it allows for flexibility in manipulating and understanding expressions.

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