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Chapter 7: Problem 47

Use the rules of exponents to simplify expression. \(\frac{8^{1 / 3}}{8^{2 / 3}}\)

Short Answer

Expert verified
\( \frac{1}{2} \)

Step by step solution

01

Identify the base

Both the numerator and the denominator have the same base: 8.
02

Apply the quotient rule of exponents

The quotient rule states \[ \frac{a^m}{a^n} = a^{m-n} \]. Here, the base is 8, so \[ \frac{8^{1/3}}{8^{2/3}} = 8^{(1/3) - (2/3)} \].
03

Simplify the exponent

Subtract the exponents: \[ \frac{1}{3} - \frac{2}{3} = -\frac{1}{3} \]. So the expression simplifies to \[ 8^{-\frac{1}{3}} \].
04

Convert to positive exponent

Using the rule \[ a^{-m} = \frac{1}{a^m} \], we get \[ 8^{-\frac{1}{3}} = \frac{1}{8^{\frac{1}{3}}} \].
05

Simplify the final expression

Since \[ 8^{\frac{1}{3}} \] is the cube root of 8, which is 2, the final simplified fraction is \[ \frac{1}{2} \].

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Key Concepts

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

simplifying exponents
Simplifying expressions involving exponents is a common task in algebra. The key is to use the rules of exponents effectively. For our original exercise, we start by identifying the base, which is 8 in both the numerator and the denominator. Since the base is the same, we can apply the quotient rule of exponents. This rule states that when you divide like bases, you subtract the exponent in the denominator from the exponent in the numerator. So, \[ \frac{8^{\frac{1}{3}}}{8^{\frac{2}{3}}} = 8^{(\frac{1}{3} - \frac{2}{3})} \] becomes \[ 8^{(-\frac{1}{3})} \]. Simplifying exponents follows this pattern of comparison and subtraction when dealing with division.
negative exponents
Negative exponents can seem tricky at first, but they follow an intuitive logic. A negative exponent indicates that you take the reciprocal of the base raised to the corresponding positive exponent. For instance, in our exercise, we simplified the expression to \[ 8^{-\frac{1}{3}} \]. According to the rule of negative exponents \[ a^{-m} = \frac{1}{a^m} \], \[ 8^{-\frac{1}{3}} = \frac{1}{8^{\frac{1}{3}}} \]. This simply means we take 1 divided by the cube root of 8. Surprisingly, turning a negative exponent into a positive one can clarify and resolve many algebraic expressions, making the math a lot easier to manage.
cube roots
A cube root is a special kind of root. It finds the number that, when multiplied by itself three times, gives you the original number. In our case, we need to find the cube root of 8. To do this, consider this equation: \[ a^3 = 8 \]. Solving for \[ a \], we find that \[ 2^3 = 8 \], so the cube root of 8 is 2. This means \[ 8^{\frac{1}{3}} = 2 \]. Going back to our expression \[ \frac{1}{8^{\frac{1}{3}}} \], it simplifies to \[ \frac{1}{2} \]. Understanding cube roots can help in various fields of math and applied sciences, making it a powerful tool in a student's toolkit.

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Most popular questions from this chapter

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