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Chapter 10: Problem 2

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

Short Answer

Expert verified
The expression \(64^{1/3}\) simplifies to 4.

Step by step solution

01

Understand the Expression

The given expression is \(64^{1/3}\). This is written in exponential form where the base is 64 and the exponent is \( \frac{1}{3} \), which indicates a cube root.
02

Convert to Radical Notation

To convert an expression in the form \(a^{m/n}\) to radical notation, we use \( \sqrt[n]{a^m} \). In this case, \(64^{1/3}\) can be written as \( \sqrt[3]{64} \), which means the cube root of 64.
03

Simplify the Radical Expression

Now, we need to find the cube root of 64. Since \(4^3 = 4 \times 4 \times 4 = 64\), the cube root of 64 is 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.

Cube Root
The cube root is a fundamental concept in mathematics. It is the inverse operation of raising a number to the third power. Just like a square root finds a number that, when multiplied by itself, gives the original number, a cube root finds a number that, when cubed, gives the original number.
For example, the cube root of 64, denoted as \( \sqrt[3]{64} \), is 4. This is because \( 4^3 = 64 \).
  • When you see the notation \( \sqrt[3]{a} \), it refers to the cube root of \( a \).
  • This concept is crucial for solving equations involving cubes and is extensively used in algebra and geometry.
Understanding cube roots helps in simplifying expressions and solving cubic equations.
Simplifying Radicals
Simplifying radicals involves rewriting a radical expression in its simplest form. The goal is to remove any perfect cube (or square, depending on the root you are dealing with) from inside the radical sign.
In the expression \( \sqrt[3]{64} \), simplifying involves recognizing that 64 can be expressed as the cube of 4, since \(4^3 = 64\).
  • This allows us to simplify \( \sqrt[3]{64} \) to the number 4, because we are effectively taking the cube root of a perfect cube.
  • Through this process, we make the expression easier to work with and understand.
Simplifying radicals is a valuable skill in algebra as it reduces complexity in equations and expressions.
Exponential Form
Exponential form is a way of expressing numbers using bases and exponents. It is compact and ideal for representing large numbers compactly.
The expression \( 64^{1/3} \) is presented in exponential form where 64 is the base and \( \frac{1}{3} \) is the exponent. This exponent indicates the cube root operation.
  • The general form \( a^{m/n} \) represents the \( n \)-th root of \( a \) raised to the \( m \)-th power. In our example, \( m = 1 \) and \( n = 3 \), simplifying to a cube root.
  • Exponential form is especially useful in scientific notation, allowing easier handling of powers and roots.
  • Converting between exponential form and radical form helps students understand different mathematical perspectives and operations.
Mastering exponential notation is crucial for higher-level math, including calculus and physics.

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Most popular questions from this chapter

The formula for the radius \(r\) of a cone with height 7 centimeters and volume \(V\) is $$ r=\sqrt{\frac{3 V}{7 \pi}} $$ Rationalize the numerator of the radical expression in this formula.

Given \(\frac{\sqrt[3]{5 y}}{\sqrt[3]{4}}\), rationalize the denominator by following parts (a) and (b). a. Multiply the numerator and denominator by \(\sqrt[3]{16}\). b. Multiply the numerator and denominator by \(\sqrt[3]{2}\). c. What can you conclude from parts (a) and (b)?

Rationalize each numerator. Assume that all variables represent positive real numbers. \(\sqrt{\frac{3}{7}}\)

Solve. $$ \sqrt[4]{2 x-9}-3=0 $$

Rationalize each numerator. Assume that all variables represent positive real numbers. \(\frac{2-\sqrt{7}}{-5}\)
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