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Simplifying Radical Expressions: The 4th Root of 64 Divided by the 4th Root of 4
In the realm of mathematics, radicals play a crucial role in representing roots of numbers. Understanding how to simplify radical expressions is fundamental to solving various mathematical problems. This blog post will delve into simplifying a specific radical expression: the 4th root of 64 divided by the 4th root of 4.
Understanding Radical Expressions
A radical expression is a mathematical expression that involves a radical symbol (√). The number under the radical symbol is called the radicand. The small number outside the radical symbol, called the index, indicates the type of root we are seeking. For instance, a square root has an index of 2, a cube root has an index of 3, and so on.
Simplifying the Expression
Let’s break down the simplification process for the expression: 4√64 / 4√4
- Find the fourth root of 64: The fourth root of 64 is 4 because 4 x 4 x 4 x 4 = 64.
- Find the fourth root of 4: The fourth root of 4 is 2 because 2 x 2 x 2 x 2 = 4.
- Divide the results: Now we have 4 / 2.
- Simplify the division: 4 / 2 = 2.
Therefore, the simplified form of 4√64 / 4√4 is 2.
Key Principles
The simplification of radical expressions relies on several fundamental principles:
- Product Rule of Radicals: n√a * n√b = n√(a * b)
- Quotient Rule of Radicals: n√a / n√b = n√(a / b)
- Simplifying Radicals: If the radicand can be factored into a perfect nth power, the radical can be simplified by taking the nth root of the perfect nth power.
Practice Problems
To solidify your understanding, try simplifying these radical expressions:
- 3√27 / 3√3
- 5√32 / 5√2
- 2√16 / 2√4
Conclusion
Simplifying radical expressions is a fundamental skill in mathematics. By applying the rules and principles discussed in this blog post, you can effectively simplify radical expressions involving division. Remember to practice and apply these concepts to enhance your problem-solving abilities in mathematics.