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Chapter 8: Problem 38

Write your answer as a power or as a product of powers. $$ (-3 a)^{5} \cdot(4 a)^{2} $$

Short Answer

Expert verified
\(-3888a^7\)

Step by step solution

01

Apply Exponent to Each Factor Inside Parentheses

The first step is to apply the exponents to each factor inside the parentheses. We have \( (-3)^5 \cdot a^5 \cdot (4)^2 \cdot a^2 \). This simplifies into \( -243 a^5 \cdot 16 a^2 \).
02

Combine Like Terms

As per the multiplication rule of exponents, when we multiply like bases, we add the exponents. Therefore, \(a^5\) multiplied by \(a^2\) is \(a^{5+2}\) or \(a^7\). Also, the numerical factors will be multiplied together to get -3888.
03

Finalize the Expression

After combining like terms, the expression simplifies to \(-3888a^7\).

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Key Concepts

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

Multiplication of Like Bases
Understanding how to multiply terms with the same base is crucial in simplifying expressions involving exponents. When we have like bases 鈥� meaning the same variable or number 鈥� raised to different powers, we add the exponents together. This process is governed by the law of exponents:
  • For example, if you have \( a^m \times a^n \), it simplifies to \( a^{m+n} \).
In the original exercise, we had \( a^5 \) and \( a^2 \). Applying the rule of multiplication for like bases, \( a^5 \cdot a^2 \) becomes \( a^{5+2} \) or \( a^7 \). This rule greatly simplifies complex expressions. Make sure to keep track of which terms share bases to avoid errors when adding exponents.
Exponentiation
Exponentiation is the mathematical operation involving numbers raised to a power. It鈥檚 a shorthand for repeated multiplication of the same factor. For instance:
  • \( x^3 \) means \( x \times x \times x \).
  • \( y^2 \) means \( y \times y \).
In the original exercise, each term inside the parentheses had to be exponentiated. This meant raising both numbers and variables to the respective powers, i.e., \( (-3)^5 \) and \( 4^2 \). An important note is the handling of negative bases: \( (-3)^5 \) results in a negative product since the exponent is odd (an even exponent would result in a positive product). Remembering these nuances protects against sign mistakes in your calculations.
Simplifying Expressions
Simplifying expressions isn't just about solving them; it's about making them as straightforward as possible. This involves combining like terms and using the properties of exponents.
  • First, simplify each component of the expression by applying the powers to numbers and variables.
  • Next, combine the numerical results and variables separately, keeping track of the signs and applying multiplication as necessary.
In our example: after exponentiating, we needed to multiply \(-243\) and \(16\) to get \(-3888\). Similarly, combining \( a^5 \) and \( a^2 \) becomes \( a^7 \). Putting these together, the final simplified expression is \(-3888a^7\). Simplifying expressions involves both following steps procedurally and occasionally re-checking each step for accuracy.

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Most popular questions from this chapter

Write your answer as a power or as a product of powers. $$ (-x)^{4}(-x)^{3}(-x)^{2} $$

Use the following information. The power generated by a windmill can be modeled by the equation \(w=0.015 s^{3},\) where \(w\) is the power measured in watts and \(s\) is the wind speed in miles per hour. Find the ratio of the power generated by a windmill when the wind speed is 20 miles per hour to the power generated when the wind speed is 10 miles per hour.

Write your answer as a power or as a product of powers. $$ \left(-2 m^{4} n^{6}\right)^{2} $$

Decide whether the ordered pair is a solution of the system. $$ \begin{aligned} &x-5 y=9\\\ &3 x+y=11 \quad(1,-4) \end{aligned} $$

Write your answer as a power or as a product of powers. $$ (3 \cdot 7)^{2} $$
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