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Chapter 0: Problem 31

Use the laws of exponents to compute the numbers. $$6^{1 / 3} \cdot 6^{2 / 3}$$

Short Answer

Expert verified
The result is 6.

Step by step solution

01

Identify the Exponential Laws

Identify which laws of exponents are applicable. Here, the product of exponents law is relevant: when you multiply two powers with the same base, you add the exponents, i.e., \[a^m \times a^n = a^{m+n}\]
02

Apply the Product of Exponents Law

Combine the exponents of the given expression, \(6^{1/3} \times 6^{2/3}\), using the product of exponents law: \[6^{1/3 + 2/3}\]
03

Add the Exponents

Add the exponents: \(1/3 + 2/3 = 3/3 = 1\) so the expression becomes: \[6^{1}\]
04

Simplify the Expression

Simplify \(6^1\) which equals 6. Therefore, \(6^{1/3} \times 6^{2/3} = 6\)

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Key Concepts

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

Product of Exponents Law
One of the core concepts in solving the given exercise is the product of exponents law. This law states that when you multiply two exponential expressions with the same base, you can add their exponents. The mathematical representation of this concept is: \(a^m \times a^n = a^{m+n}\). To apply this law, ensure that the base of both exponent expressions is identical. For example, in the exercise \(6^{1/3} \times 6^{2/3}\), both exponents share the same base, which is 6. Therefore, you can add the exponents \((1/3 + 2/3)\) and then simplify. Remember, this law only applies when the bases are the same, making our task simpler by reducing the problem to addition of the exponents.
Simplifying Exponents
After applying the product of exponents law, you need to simplify the exponents. In the given exercise, after combining the exponents \(6^{1/3 + 2/3}\), your task is to simplify the sum of the exponents: \(1/3 + 2/3 = 3/3 = 1\). Now, your expression has turned from a fraction to a whole number \(6^1\). Simplifying exponents is crucial as it transforms complex exponent rules into easier arithmetic. Whenever you encounter fractions as exponents, try converting them into simpler terms or whole numbers, making the expression straightforward to handle.
Basic Algebra
Understanding laws of exponents is a valuable part of basic algebra. Algebra involves various rules and properties that simplify expressions and solve equations effectively. Whether it's the product law, quotient law, or power of a power law, exponents play a crucial role. The given exercise can be broken down using simple algebraic concepts:
  • Identifying the correct law to apply
  • Simplifying exponents through basic arithmetic
So, when you see expressions like \(6^{1/3} \times 6^{2/3}\), use your knowledge of algebraic principles. First, decide which exponent rule fits best, apply it, and finally, simplify the expression. Mastery over these fundamental algebra concepts will ease your way through more complex algebraic problems.

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Most popular questions from this chapter

Let \(f(x)=\frac{x}{x-2}, g(x)=\frac{5-x}{5+x},\) and \(h(x)=\frac{x+1}{3 x-1} .\) Express the following as rational functions. $$\frac{g(x+5)}{f(x+5)}$$

Evaluate \(f(4)\). $$f(x)=x^{3 / 2}$$

Convert the numbers from graphing calculator form to standard form (that is, without E). $$5 E-5$$

Use the laws of exponents to simplify the algebraic expressions. Your answer should not involve parentheses or negative exponents. $$\sqrt{x}\left(\frac{1}{4 x}\right)^{5 / 2}$$

Find a good window setting for the graph of the function. The graph should show all the zeros of the polynomial. $$f(x)=3 x^{3}+52 x^{2}-12 x-12$$
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