Problem 127 - Elementary and Intermediate Algebra

Question

Simplify each expression. All variables represent positive real numbers. $$ \sqrt[4]{25 b^{2}} $$

Step-by-Step Solution

Answer

The simplified expression is \( \sqrt{5b} \).

Step 1: Understand the Expression

We begin with simplifying the expression \( \sqrt[4]{25 b^{2}} \). This is a fourth root, which means we are looking to simplify \( 25 b^2 \) under the fourth root.

Step 2: Prime Factorization of 25

Recognize that \( 25 \) can be written as a power of 5: \( 25 = 5^2 \). This allows us to rewrite the expression as \( \sqrt[4]{5^2 b^2} \).

Step 3: Use the Property of Exponents and Fourth Root

We can apply the property \( \sqrt[n]{x^m} = x^{m/n} \). Using this property: \( \sqrt[4]{5^2} = (5^2)^{1/4} = 5^{2/4} = 5^{1/2} \) and \( \sqrt[4]{b^2} = (b^2)^{1/4} = b^{2/4} = b^{1/2} \).

Step 4: Combine the Results

Combine the results of the fourth roots: \( \sqrt[4]{25 b^{2}} = 5^{1/2} \, b^{1/2} \). This can be more simply represented as \( (5b)^{1/2} \), which is equivalent to \( \sqrt{5b} \).

Key Concepts

Understanding the Fourth Root

When dealing with the fourth root, we are exploring the concept of finding a number which, when used four times as a factor, equals the original number. In mathematical terms, if you have a number \( x \), the fourth root of \( x \), denoted as \( \sqrt[4]{x} \), is the number \( y \) such that \( y^4 = x \). For instance, the fourth root of 16 is 2, because raising 2 to the power of four gives us 16.

The process of determining the fourth root can often involve converting that number into a form that makes the calculation easier. It often includes breaking down a number into its prime factors and then applying the concept of exponents to simplify it.

Exponent Properties Simplified

Exponent properties are a crucial tool in simplifying expressions, especially when dealing with roots. The fundamental property to remember is \( \sqrt[n]{x^m} = x^{m/n} \). This transformation helps in simplifying roots to more manageable exponent expressions.

For example, when you have \( (x^m)^{1/n} \), it means you are distributing the root across the exponent, dividing the power by the root, resulting in \( x^{m/n} \). Let's look at some core exponent properties:

• Product of Powers: \( a^m \cdot a^n = a^{m+n} \)
• Power of a Power: \( (a^m)^n = a^{m\cdot n} \)
• Power of a Product: \( (ab)^n = a^n \cdot b^n \)
• These properties simplify calculations and make it easier to work with long and complex expressions.

By understanding and applying these properties, one can change between exponential forms and root forms, leading to a simplified expression.

Breaking Down Prime Factorization

Prime factorization involves breaking a number down into its basic building blocks, which are prime numbers. A prime number is a number greater than 1 that has no divisors other than 1 and itself.

This process is essential in algebraic simplification because it allows us to express a number as the product of primes, making it easier to understand and simplify under roots or exponents.

• For example, 25 can be broken down into its prime factors as \( 5 \times 5 \) or \( 5^2 \).
• This was a critical step in the original problem, allowing us to express the fourth root as \( \sqrt[4]{5^2} \), which simplifies using the exponent property \( 5^{1/2} \).

Prime factorization is not only a method for simplifying roots and exponents but also a crucial foundational skill within mathematics, aiding in greater understanding and manipulation of various expressions.