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Chapter 0: Problem 78

Simplify. $$ 8^{2 / 3} $$

Short Answer

Expert verified
The simplified form of \( 8^{2/3} \) is 4.

Step by step solution

01

Understand the Expression

The expression to simplify is \( 8^{2/3} \). This is a fractional exponent, which means we need to understand it as a root and a power.
02

Convert Exponent to Radical Form

A fractional exponent like \( a^{m/n} \) can be understood as either \( (a^m)^{1/n} \) or \( (a^{1/n})^m \). In our case \( 8^{2/3} \) can be rewritten as \( (8^{1/3})^2 \).
03

Find the Cube Root

The cube root of 8 is \( 2 \), because \( 2^3 = 8 \). So, \( 8^{1/3} = 2 \).
04

Square the Result

Now we square the result of the cube root: \( 2^2 = 4 \). Therefore, \( (8^{1/3})^2 = 4 \).

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Key Concepts

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

Simplifying Exponents
Simplifying exponents is like cleaning up a math expression to make it easier to understand or work with. Often, this involves working with fractional exponents. A fractional exponent can look confusing, but it simply represents a combination of taking roots and raising to a power.
  • The numerator of the fractional exponent tells you the power to which the base is raised.
  • The denominator indicates the root you need to find.
For example, in the expression \(8^{2/3}\), the number 3 in the fraction's denominator indicates that you'll take the cube root of 8. The number 2 in the numerator tells you to square the result afterward. So, by understanding both parts separately, you can simplify the expression effectively.
Radical Expressions
A radical expression involves roots, like the square root or the cube root. When you see a fractional exponent such as \(a^{m/n}\), it tells you to use a radical expression.
In the case of \(8^{2/3}\), you can rewrite this as a radical by understanding it as \((8^{1/3})^2\). Here:
  • \(8^{1/3}\) means "the cube root of 8".
  • This radical expression is easier to manage and calculate step-by-step.
Radical expressions are just another way of expressing roots that make use of both roots and powers.
Cube Root
The cube root is a special type of root that given a number, finds another number which when multiplied by itself three times gives the original number. In mathematical terms, if \(x = y^3\), then \(y\) is the cube root of \(x\).
For the number 8, its cube root is 2 because \(2^3 = 8\). This is denoted as \(8^{1/3} = 2\).
  • The cube root is particularly useful in simplifying expressions with exponents where the base is a perfect cube.
  • Cube roots are often simpler to handle than higher-order roots like the fourth or fifth roots.
In expressions with fractional exponents, the cube root keeps calculations simple and manageable.
Exponentiation
Exponentiation is all about repeating multiplication. It tells you how many times a number, called the base, is multiplied by itself.
In the context of fractional exponents, exponentiation combines with roots. After finding a root, you "exponentiate" by raising the result to a certain power which is indicated by the numerator of the fractional exponent.
For example, after calculating the cube root of 8 as 2, to simplify \(8^{2/3}\), you take 2 and raise it to the power of 2: \(2^2 = 4\). This offers:
  • A concise way of simplifying expressions.
  • Enables solving equations that are otherwise tough to evaluate directly.
The power of exponentiation lies in its simplicity and ability to resolve complex power and root problems efficiently.

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