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Chapter 7: Problem 72

Simplify. Use absolute-value notation when necessary. $$ \sqrt[1976]{(2 a+b)^{1976}} $$

Short Answer

Expert verified
|2a + b|

Step by step solution

01

Identify the expression

The given expression is \( \sqrt[1976]{(2a + b)^{1976}} \).
02

Apply the property of radicals

Using the property \( \sqrt[n]{x^n} = |x| \) for any integer \( n \) and real number \( x \), where \( n \) is even, we can simplify the expression.
03

Simplify the expression

Since \( n = 1976 \) is an even number, applying the radical property, we get: \[ \sqrt[1976]{(2a + b)^{1976}} = |2a + b| \]
04

Final answer

Thus, the simplified form of the given expression is \( |2a + b| \).

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Key Concepts

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

Absolute Value
Absolute value is a fundamental concept in mathematics. It measures the distance of a number from zero on the number line, without considering which direction (positive or negative) the number is.
For any real number, the absolute value is denoted by vertical bars. For example, the absolute value of -3 is written as \(|-3|\), which equals 3.

Here are some key points to understand about absolute values:
  • It always results in a non-negative number.
  • It is essential when dealing with radical expressions and simplifying them.
Another way to think of absolute value is by considering scenarios where direction doesn't matter, such as distance or magnitude. In our exercise, the absolute value helps ensure our answer is non-negative since the radical's power (1976) is even.
Property of Radicals
The property of radicals simplifies the process of dealing with complex expressions involving roots. This property states that if you have an expression like \( \sqrt[n]{x^n}\), it simplifies to \(|x|\) when \ is an even integer.

The key points of this property include:
  • It only applies when \ is even.
  • It helps in reducing the radical expression to an absolute value.
This property is helpful, particularly when you have a high power or root, as it significantly simplifies the problem. In our exercise, we applied this property to reduce \(\root[1976]{(2a + b)^{1976}}\) to \(|2a + b|\).
Even Integers
Understanding even integers is crucial in simplifying radical expressions. An even integer is any integer divisible by 2, such as 2, 4, 6, 8, and so on.

The relevance in our exercise is connected to the property of radicals. When the radical's power \ is an even integer, it guarantees specific simplifications involving absolute values. Here's why it matters:
  • Even powers eliminate the possibility of negative results (because raising any real number to an even power always gives a non-negative result).
  • Allows us to apply the property \(\root[n]{x^n} = |x|\) straightforwardly.
In our specific case, since 1976 is an even number, it enabled us to simplify \(\root[1976]{(2a + b)^{1976}}\) directly to \(|2a + b|\), ensuring a clear and concise answer.

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Most popular questions from this chapter

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