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Chapter 6: Problem 7

In Exercises 3鈥�12, simplify the expression. $$ \left(5 e^{7 x}\right)^4 $$

Short Answer

Expert verified
The simplified expression for \( (5e^{7x})^4 \) is \( 625e^{28x} \)

Step by step solution

01

Apply the Power of a Power rule

According to the power of a power rule, we can multiply the exponents when a power is raised to another power. Here, \( (5e^{7x})^4 \) can be rewritten as \( 5^4 \times (e^{7x})^4 \)
02

Simplify the expression

First, simplify \( 5^4 \) to get 625. Then, apply the power of power rule to \( (e^{7x})^4 \), which results in \( e^{7x \times 4} \). The expression thus far is \( 625e^{28x} \)
03

Final Answer

Therefore, the simplified form of the given expression \( (5e^{7x})^4 \) is \( 625e^{28x} \)

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Key Concepts

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

Exponent Rules
Understanding the rules of exponents is crucial in algebra, as they allow us to manipulate and simplify expressions effectively. Here are some key exponent rules to remember:
  • Product of Powers Rule: If you have the same base, you add the exponents: \( a^m \times a^n = a^{m+n} \)
  • Quotient of Powers Rule: Again, with the same base, subtract the exponents: \( \frac{a^m}{a^n} = a^{m-n} \)
  • Zero Exponent Rule: Any base except zero, raised to the power of zero, equals one: \( a^0 = 1 \)
  • Negative Exponent Rule: A negative exponent indicates a reciprocal: \( a^{-n} = \frac{1}{a^n} \)
By using these rules, you can simplify complex algebraic expressions, making problems more manageable. They serve as a foundation for further operations, like those involved in the power of a power rule.
Power of a Power Rule
The power of a power rule is a powerful tool for simplifying expressions where an exponent is raised to another exponent. This rule states: The power of a power rule allows us to multiply the exponents while keeping the base unchanged.
For example, if you have \( (a^m)^n \), you can simplify it to \( a^{m \times n} \).To see this rule in action, consider the expression \( (5e^{7x})^4 \). Using the power of a power rule:
  • The number 5 is raised to the fourth power, resulting in \( 5^4 \).
  • For the term \( e^{7x} \), applying the rule means multiplying the exponents: \( e^{7x \times 4} = e^{28x} \).
This rule helps streamline calculations and is particularly useful in dealing with exponential growth and decay problems in math.
Simplifying Expressions
Simplifying expressions is about reducing them to their most concise and understandable form. This process involves applying mathematical operations strategically, especially using exponent rules.
Let's consider the expression \( (5e^{7x})^4 \):
  • First, apply the power of a power rule to split and simplify each part: \( 5^4 \) produces 625.
  • Then, simplify the exponential term by multiplying its exponents: \( e^{28x} \).
These steps help to transform a seemingly complex expression into one that's much easier to work with: \( 625e^{28x} \). Always aim to break down expressions using fundamental algebraic rules, which will provide clarity and make solving mathematical problems much more straightforward. This process not only enhances your skills but also deepens your understanding of mathematical concepts.

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