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Chapter 5: Problem 45

Use the power rule and the power of a product or quotient rule to simplify each expression. See Examples 6 through 8 . $$ \left(2 a^{5}\right)^{3} $$

Short Answer

Expert verified
The simplified expression is \(8a^{15}\).

Step by step solution

01

Apply the Power Rule

The power rule states that when raising a power to another power, you multiply the exponents. The expression is \((2a^5)^3\). Here, the exponent of \(a\) is 5 and it is raised to the power of 3. We apply the power rule for \(a\): \(a^{5 \times 3}\).
02

Simplify the Exponents

Using the multiplication result from Step 1, calculate the new exponent for \(a\). That gives us \(a^{15}\).
03

Apply the Power Rule to the Constant

The coefficient 2 is also raised to the power of 3. According to the power rule for the coefficient, compute \(2^3\).
04

Simplify the Entire Expression

Calculate \(2^3 = 8\). Now combine this with the simplified variable component to get the final result: \(8a^{15}\).

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Key Concepts

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

Exponentiation
Exponentiation is a mathematical operation that involves raising a number, called the base, to a certain power, which is the exponent. In the context of our problem, the expression \( (2a^5)^3 \) involves exponentiation. Here, the base is the product \( 2a^5 \), and it is raised to the power of 3. This means we need to multiply \( 2a^5 \) by itself three times.
Understanding exponentiation is crucial for simplifying algebraic expressions, especially when dealing with powers of powers. Always remember: the exponent tells us how many times to use the base in a multiplication.
Algebraic Expression Simplification
Algebraic expression simplification involves making an expression easier to work with by using algebraic rules. In our example, \( (2a^5)^3 \), simplification is primarily focused on reducing the expression to its most basic components.
Steps to consider include:
  • Applying rules like the power rule and power of a product rule to combine and simplify terms.
  • Multiplying exponents when a power is raised to another power.
  • Calculating powers of coefficients (numbers that multiply variables).
Implementing these steps correctly yields a cleaner, more manageable expression, enabling us to see the relationships between different parts of an equation.
Power of a Product Rule
The Power of a Product Rule is a fundamental rule in algebra that helps simplify expressions like \( (ab)^n \), where \(a\) and \(b\) are different bases and \(n\) is the exponent. This rule states that \((ab)^n = a^n b^n\). This rule can be extended to any number of terms within a set of parentheses raised to a power.
In our problem, \( (2a^5)^3 \) is a classic application. Using this rule:
  • Raise each factor in the product to the given power: \((2^3)(a^{5\times 3})\).
  • This leads directly to \(8a^{15}\) through proper exponentiation.
By applying the Power of a Product Rule, we effectively break down complex expressions into simpler parts, enabling easier manipulation and solutions.

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Most popular questions from this chapter

Multiply. See Section 5.1 $$ \left(12 x^{3}\right)\left(-x^{5}\right) $$

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See the Concept Checks in this section. $$5 a+6 a$$

Evaluate each expression using exponential rules. Write each result in standard notation. See Example 7. $$ \left(5 \times 10^{6}\right)\left(4 \times 10^{-8}\right) $$

Multiply. $$(5 x+4)\left(x^{2}-x+4\right)$$
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