Problem 54 - Algebra 2

Question

Simplify each number. $$64^{3.5}$$

Step-by-Step Solution

Answer

Consequently, \(64^{3.5}\) simplifies to 2,097,152.

Step 1: Understand the Expression

Firstly, note that the expression, \(64^{3.5}\), is the same as \(64^{7/2}\). This is because 3.5 as a fraction is 7/2. Therefore, the expression can be written as \((64^{1/2})^7\), that signifies the square root of 64 to the power of 7.

Step 2: Simplify Inner Brackets

Here, it's essential to calculate the square root of 64 which equals 8. Thus, it becomes \(8^7\)

Step 3: Calculate the Power of 7

Following, compute \(8^7\), which equals 2,097,152.

Key Concepts

Fractional Exponents

Fractional exponents are a way to represent roots and powers in mathematics. They allow us to express roots as exponents, making calculations more straightforward.

• Understanding: A fractional exponent such as \(3.5\) can be converted into a fraction \(\frac{7}{2}\), where the numerator is the power and the denominator is the root.
• Usage: This means \(64^{3.5}\) is the same as \(64^{\frac{7}{2}}\), indicating the square root of 64, raised to the seventh power.
• Conversion Advantage: The use of fractional exponents simplifies the process of dealing with roots and powers, allowing for more streamlined calculations.

Applying fractional exponents can help you work with more complex mathematical expressions by breaking them down into easier steps.

Roots and Radicals

Roots and radicals are essential parts of expressions involving fractional exponents. They describe the process of finding a number that, when raised to a specified power, gives the original number.

• Root Definition: The root of a number is essentially its inverse power. For example, the square root of a number \(x\) is expressed as \(x^{1/2}\).
• Radical Notation: It is often represented with the radical symbol \(\sqrt{}\), which denotes the principle that there is some number that squares back to the original value.
• Calculating Example: In the exercise, we find \(64^{1/2} = 8\). This translates to finding which number, when squared, results in 64.

Mastering the concept of radicals is essential for simplifying expressions that involve roots, making complex calculations manageable.

Simplifying Expressions

Simplifying expressions is the process of breaking down complex expressions into their simplest form to make calculations and comprehension easier.

• Identify Parts: Start by breaking down the expression into understandable parts, like recognizing \(3.5\) as \(\frac{7}{2}\).
• Step-by-Step Approach: First, simplify any inner mathematical operations such as square roots. With \(64^{1/2} = 8\), identify intermediate results before executing further calculations.
• Final Calculation: Once we have simplified the intermediate steps, compute the full power: \(8^7 = 2,097,152\).

By understanding and applying these steps, even seemingly complicated problems become straightforward and approachable.