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Chapter 10: Problem 2

Use radical notation to write each expression. Simplify if possible. $$64^{1 / 3}$$

Short Answer

Expert verified
\(64^{1/3} = \sqrt[3]{64} = 4\).

Step by step solution

01

Understand Radical Notation

The expression given is in exponential form, specifically of the type involving fractional exponents. In radical notation, the exponent fraction \(\frac{1}{n}\) corresponds to the \(n\)-th root of the base. So here, \(64^{1/3}\) means the cube root of 64.
02

Convert to Radical Notation

To convert \(64^{1/3}\) to radical notation, write it as the cube root of 64, which is \(\sqrt[3]{64}\). The base 64 is placed under the radical symbol, and the denominator of the exponent, 3, is the root index.
03

Simplify the Radical

Identify the cube root of 64. Since \(64 = 4^3\), the cube root of 64 simplifies to 4. Thus, \(\sqrt[3]{64} = 4\).

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Key Concepts

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

Fractional Exponents
Fractional exponents are a unique way to represent expressions that involve roots. Normally, when you see an exponent, it suggests how many times to multiply a number by itself. However, a fractional exponent has a different role.
  • The numerator of the fraction denotes the power to which the base is raised.
  • The denominator tells you which root to take of the base.
For example, in the expression \(64^{1/3}\), the exponent is \(\frac{1}{3}\). Hence, the base 64 is raised to the power of 1 (which means it remains 64) and then the cube root is taken because of the denominator 3. Understanding this relationship is crucial for converting between exponential and radical notations, making it easier to simplify expressions.
Cube Root
A cube root is a special type of root used to determine what number, when multiplied by itself three times, equals the given number. Think of it as reversing the process of cubing a number. For instance, if we take the cube of 4, we get 64 (because \(4 \times 4 \times 4 = 64\)). To find the cube root, we ask what number, when multiplied three times, gives 64. The answer is 4, so the cube root of 64 is 4.
  • The cube root is represented using the radical symbol with a small three, like this: \(\sqrt[3]{64}\).
  • It is used primarily when dealing with fractional exponents and radical notation.
Mastering the concept of cube roots helps in evaluating expressions with fractional exponents, simplifying them to a more understandable form.
Simplifying Radicals
Simplifying radicals is the process of breaking down a radical expression into its simplest form. This often involves identifying perfect powers within the number under the radical sign and simplifying them out. When simplifying a cube root, look for any numbers that can be expressed as a perfect cube. For \(\sqrt[3]{64}\), recognizing that 64 is \(4^3\) allows for simplification directly to 4.Steps to simplify include:
  • Identifying which number cubed results in the original number.
  • Expressing the original number as a power of this base.
  • Applying the radical to arrive at the simplest form.
Using these steps ensures you always find the most reduced version of a radical expression, which simplifies calculations and makes understanding the expression easier.

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Most popular questions from this chapter

Identify the domain and then graph each function. $$f(x)=\sqrt{x+1}$$ use the following table. $$\begin{array}{|c|c|} \hline x & {f(x)} \\ \hline-1 & {} \\ \hline 0 & {} \\ \hline 3 & {} \\ \hline 8 & {} \\ \hline \end{array}$$

Simplify each exponential expression. $$\left(-14 a^{5} b c^{2}\right)\left(2 a b c^{4}\right)$$

Which of the following are not real numbers? $$\sqrt[3]{-17}$$

Factor the common factor from the given expression. $$ x^{-1 / 3} ; 5 x^{-1 / 3}+x^{2 / 3} $$

Factor the common factor from the given expression. $$ x^{2 / 7} ; x^{3 / 7}-2 x^{2 / 7} $$
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