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Chapter 7: Problem 54

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

Short Answer

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

Step by step solution

01

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.
02

Simplify Inner Brackets

Here, it's essential to calculate the square root of 64 which equals 8. Thus, it becomes \(8^7\)
03

Calculate the Power of 7

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

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Key Concepts

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

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.

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Most popular questions from this chapter

Graph. Find the domain and the range of each function. \(y=\sqrt{x}-6\)

The time \(t\) in seconds for a trapeze to complete one full cycle is given by the function \(t=1.11 \sqrt{\ell}\) , where \(\ell\) is the length of the trapeze in feet. a. Graph the equation on your calculator. Make a sketch of the graph. b. How long is a full cycle if the trapeze is 15 ft. long? 30 ft. long?

Graph. Find the domain and the range of each function. \(y=-1-\sqrt{4 x+20}\)

Solve each square root equation by graphing. Round the answer to the nearest hundredth if necessary. If there is no solution, explain why. \(2.5 \sqrt{2 x-1.3}=-1\)

List all possible rational roots for each equation. Then use the Rational Root Theorem to find each root. $$ x^{3}-6 x^{2}+11 x-6=0 $$
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