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Chapter 6: Problem 10

In Exercises 3鈥�12, simplify the expression. $$ \sqrt[3]{8 e^{12 x}} $$

Short Answer

Expert verified
So, the simplified form of \(\sqrt[3]{8 e^{12 x}}\) is \(2e^{4x}\).

Step by step solution

01

Break down the cube root

The cube root can be identified as \(\sqrt[3]{a}\) where a is the number inside the cube root. In the exercise, we have cube root of 8 which is equal to 2. So, \(\sqrt[3]{8} = 2\).
02

Simplifying the exponent within the cube root

Now, let's simplify the exponential part, which is \(e^{12x}\). According to the rule of exponentiation, \(e^{12x}\) under a cube root is the same as \(e^{(12x/3)} = e^{4x}\). So, \(\sqrt[3]{e^{12x}} = e^{4x}\).
03

Combining the results

Combining the results from steps 1 and 2, we get \(2e^{4x}\). So, \(\sqrt[3]{8 e^{12 x}} = 2e^{4x}\).

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Key Concepts

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

Cube Roots
A cube root is a special kind of root. It answers the question: What number multiplied by itself twice gives the original number? For example, the cube root of 8 is 2. This is because when you multiply 2 by itself three times, you get 8 (since \( 2 \times 2 \times 2 = 8 \)).

When simplifying cube roots, always look for perfect cubes, which are numbers that can be expressed as a whole number to the power of three. Here are some examples of perfect cubes:
  • Cubing 1 gives 1, so \( \sqrt[3]{1} = 1 \).
  • Cubing 2 gives 8, so \( \sqrt[3]{8} = 2 \).
  • Cubing 3 gives 27, so \( \sqrt[3]{27} = 3 \).
  • Cubing 4 gives 64, so \( \sqrt[3]{64} = 4 \).
Simplifying cube roots can make calculations more manageable. It involves recognizing those patterns and using them to simplify expressions inside a cube root.
Exponentiation
Exponentiation involves raising numbers to certain powers, which is used to express repeated multiplication easily. For example, \( e^{12x} \) is an exponential expression where \( e \) (Euler's number, approximately 2.718) is raised to the power of \( 12x \).

Key exponentiation rules include:
  • Product Rule: If the bases are the same, you can add the exponents: \( a^m \times a^n = a^{m+n} \).
  • Power Rule: When raising an exponential expression to a power, multiply the exponents: \( (a^m)^n = a^{m \cdot n} \).
  • Quotient Rule: If the bases are the same, you can subtract the exponents: \( a^m / a^n = a^{m-n} \).
When dealing with cube roots, the Power Rule helps simplify the process. For example, \( e^{12x} \) under a cube root translates to dividing the exponent by 3, simplifying to \( e^{4x} \). Understanding these rules makes working with exponential expressions much easier.
Algebraic Simplification
Algebraic simplification is the process of making an expression easier to work with. The ultimate goal is to reduce complexity while maintaining equivalence. In the provided exercise, simplification involved breaking down multiple components into a more convenient form.

Steps to algebraic simplification may include:
  • Applying arithmetic operations like addition, subtraction, multiplication, and division.
  • Combining like terms to reduce the number of terms.
  • Utilizing mathematical identities or properties, such as distributive, associative, or commutative properties.
  • Recognizing and simplifying root functions and exponential expressions, as done in identifying \( \sqrt[3]{8} = 2 \) and \( \sqrt[3]{e^{12x}} = e^{4x} \).
By systematically applying these strategies, complex expressions become easier to handle and understand, facilitating further mathematical manipulation or application.

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