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Chapter 7: Problem 15

Evaluate each exponential. $$ 729^{1 / 3} $$

Short Answer

Expert verified
The value of \( 729^{1 / 3} \) is 9.

Step by step solution

01

Understand the Problem

The problem asks to evaluate the exponential expression \( 729^{1 / 3} \). This involves finding the cube root of 729.
02

Recognize the Exponential Fraction

The expression \( 729^{1 / 3} \) can be interpreted as the cube root of 729, which means we need to find a number that, when raised to the power of 3, equals 729.
03

Find the Cube Root

To find the cube root, consider the fact that \( 729 = 9 \times 9 \times 9 \). Hence, \( 9^3 = 729 \). Therefore, \( 729^{1/3} = 9 \).
04

Confirm the Calculation

Verify the result by calculating \( 9^3 \). If \( 9^3 = 729 \), then the cube root of 729 is indeed 9.

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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 of a number is a value that, when multiplied by itself three times, gives the original number.
For example, the cube root of 27 is 3 because \( 3 \times 3 \times 3 = 27 \).
In notation, the cube root of a number x is represented as \( \sqrt[3]{x} \. \) In some cases, this can be written using exponentiation as \( x^{1/3} \).
Exponentiation
Exponentiation is a mathematical operation involving two numbers, the base and the exponent.
The exponent indicates how many times the base is multiplied by itself.
For instance, in the expression \( 2^3 \), 2 is the base and 3 is the exponent, representing \( 2 \times 2 \times 2 = 8 \). Exponentiation extends beyond whole numbers; fractions can also be used.
For example, \( 729^{1/3} \) represents the cube root of 729.
Radicals in Algebra
Radicals in algebra denote roots of numbers using radical symbols.
The form of a root can vary, such as a square root \( \sqrt{x} \) (for square roots) or a cube root \( \sqrt[3]{x} \) (for cube roots).
These symbols simplify repeated multiplication into a more accessible notation.
For example, the cube root of 8 can be written as \( \sqrt[3]{8} \) or \( 8^{1/3} \), both representing \( 2 \). Understanding radicals is crucial for solving exponential equations and simplifying expressions.

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Most popular questions from this chapter

Write each quotient in lowest terms. Assume that all variables represent positive real numbers. $$ \frac{30+20 \sqrt{6}}{10} $$

Rationalize each denominator. Assume that all variables represent positive real numbers and that no denominators are 0. $$ \frac{2}{3 \sqrt{5}+2 \sqrt{3}} $$

Rationalize each numerator. Assume that all variables represent positive real numbers. $$ \frac{2 \sqrt{5}-3}{2} $$

Find the distance between each pair of points. (4.7,2.3) and (1.7,-1.7)

Rationalize each denominator. Assume that all variables represent positive real numbers and that no denominators are 0. $$ \frac{r-9}{\sqrt{r}-3} $$
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