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Chapter 10: Problem 32

Write an equivalent expression using exponential notation. $$ \sqrt[3]{19} $$

Short Answer

Expert verified
The equivalent expression is \( 19^{\frac{1}{3}} \).

Step by step solution

01

- Understanding the Cube Root

The expression is written as a cube root: \(\root {3} \ 19 \). A cube root is another way of writing an expression that is raised to the power of \(\frac{1}{3}\).
02

- Converting to Exponential Notation

To write the cube root in exponential notation, use the rule that \( \root {n} \ a \) can be written as \( a ^ \frac{1}{n} \). Therefore, \(\root {3} \ 19 \) becomes \(19^{\frac{1}{3}} \).
03

- Final Equivalent Expression

The cube root of 19 in exponential notation is \( 19^{\frac{1}{3}} \).

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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 number that, when multiplied by itself three times, gives the original number. For example, the cube root of 8 is 2, because 2 * 2 * 2 equals 8. The symbol for cube root is \(\root {3} \ \ \ \ \ \), and it means the same as raising a number to the power of 1/3. So, \( \sqrt[3]{19} = 19^{1/3} \).
This notation is useful in algebra and makes many calculations easier.
When solving problems, remember:
  • The cube root gives a smaller number that can recreate the original through multiplication.
  • It is always the number that you multiply by itself three times.
  • The cube root of a number can be positive or negative since both can multiply to a positive result (e.g., (-2) * (-2) * (-2) = -8).
Exponents
Exponents are a shorthand way of expressing repeated multiplication of the same number by itself. When we use exponents, we write them as superscripts. For example:
\(2^3=2*2*2=8\).
In the context of cube roots, exponents are used to express how many times a number is multiplied by itself to reach the cube. When you see \( x^{\frac{1}{3}} \), it means you're looking for the cube root of x.
Here are some important rules:
  • \frac{1}{n} = The root of the number. So \( a^{\frac{1}{n}}= \root {n} a = n \)
  • Exponents follow specific rules when you multiply, divide, or raise other exponents.
  • You can convert roots into fractional exponents to simplify calculations.
Understanding these principles will make it easier to handle algebraic expressions that involve roots and powers.
Algebra
Algebra is a way of solving for unknown values in equations. It uses symbols, usually letters, to represent numbers whose values are not yet known.
To work with algebra, especially with cube roots and exponents, remember these key points:
  • Equations can often be simplified by converting roots to exponents.
  • Remember \((a^m)^n = a^{m*n}\) to manage complex expressions.
  • The laws of exponents and roots help solve equations systematically.
For instance, solving \( \sqrt[3]{19} = 19^{\frac{1}{3}} \) demonstrates how converting between roots and exponents simplifies the expression. This makes equations easier to handle, whether you're solving for \ x \, balancing both sides, or transforming a formula.

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