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Chapter 0: Problem 20

Compute the numbers. $$(27)^{2 / 3}$$

Short Answer

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Step by step solution

01

Identify the Base and Exponent

The given expression is \( (27)^{2/3} \). Here, 27 is the base and \( \frac{2}{3} \) is the exponent.
02

Rewrite the Exponent

An exponent of the form \( \frac{m}{n} \) means taking the n-th root of the base and then raising it to the m-th power. So, \( (27)^{2/3} \) can be rewritten as \( (\sqrt[3]{27})^2 \).
03

Find the Cube Root

Calculate the cube root of 27. \( \sqrt[3]{27} = 3 \), since \( 3^3 = 27 \).
04

Raise to the Power

Now, take the result from the previous step and raise it to the second power: \( 3^2 = 9 \).

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Key Concepts

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

Base and Exponent
In the expression \( (27)^{2/3} \), the number 27 is called the 'base' and \( \frac{2}{3} \) is called the 'exponent'.
The base is the number that is being multiplied.
The exponent tells you how many times to use the base in a multiplication.
For fractional exponents like \( \frac{2}{3} \), it means you have to deal with roots as well as powers.
Always identify the base and exponent when working with exponents to make calculations easier!
This will help you transform and simplify the expression correctly.
Cube Root
A cube root finds a number which when multiplied by itself three times gives the base.
In the example \( \sqrt[3]{27} \), the number 3 is multiplied by itself three times to get 27.
Mathematically, this is expressed as:
\[ \ \sqrt[3]{27} = 3 \]
So, \( 3^3 = 27 \).
To solve our problem, you first find the cube root (which is 3), and then proceed to the next step.
Power Rules
Power rules are essential for handling expressions with exponents.
For a fractional exponent, such as \( \frac{m}{n} \), it鈥檚 important to remember that it means taking the n-th root first, then raising to the m-th power:
\[ (a^{m/n}) = (\sqrt[n]{a})^m \]
So using this rule on our expression: \( (27)^{2/3} = (\sqrt[3]{27})^2 \).
First, find the cube root which is 3 and then raise it to the power of 2.
This means:
\[ 3^2 = 9 \]
By following these power rules, the calculations become systematic and easier to handle!

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