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Chapter 1: Problem 102

Raising a Product or a Quotient to a Power $$ (-5 a b)^{3} $$

Short Answer

Expert verified
-125a^3b^3

Step by step solution

01

Determine the Expression

We need to raise the entire expression \( -5ab \) to the power of 3: \( (-5ab)^3 \)
02

Distribute the Exponent

Apply the exponent 3 to each factor inside the parenthesis: \( (-5)^3, a^3, b^3 \)
03

Calculate Each Part

Compute \( (-5)^3 \) which equals \( -125 \): \[ (-5)^3 = -5 \times -5 \times -5 = -125 \]
04

Combine Results

Multiply the computed values back together: \(-125 \cdot a^3 \cdot b^3 \). The final expression is \(-125a^3b^3 \)

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Key Concepts

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

raising a product to a power
When raising a product to a power, every factor inside the parentheses must be raised to that power individually. This rule applies no matter how many factors you have in your product.
In the exercise, we start with the product \(-5ab\) and need to raise it to the power of 3.
Therefore, we distribute the exponent to each factor inside the parentheses:
  • \(-5^3\)
  • \(a^3\)
  • \(b^3\)
    • By distributing the exponent to each part, we simplify the calculation and apply the exponent rule accurately.
exponentiation
Exponentiation is a mathematical operation that involves raising a number or expression to a specified power. This means multiplying the number by itself for the given number of times.
In the exercise, we raise \(-5 ab\) to the power of 3. Specifically, we calculate \((-5)^3\), which equals \-125\.
The calculation follows these steps:
  • Multiply \(-5\) by itself: \(-5 \times -5 = 25\)
  • Then multiply by \(-5\) again: \(25 \times -5 = -125\)
    • Thus, \((-5)^3 = -125\). Each variable inside the parentheses (\
multiplication of variables
When it comes to multiplication of variables, each variable follows the rules of exponentiation when raised to a power. In the exercise, we have to raise both \(a\) and \(b\) to the 3rd power separately.
This translates to \(a^3 \) and \(b^3 \).
Let's break it down:
  • Exponentiation is applied to each variable: \(a^3\)
  • Similarly, \(b^3\) is calculated independently
    • Finally, we combine all parts to form the final expression: \(-125 a^3 b^3 \) Combining variables raised to power follows this important point: the separate factors, whether constants or variables, are independently raised to the specified power, maintaining the initial arithmetic integrity of the expression.

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