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Chapter 8: Problem 84

Use the definition of exponents to simplify each expression. \((0.5)^{4}\)

Short Answer

Expert verified
\((0.5)^4 = 0.0625\).

Step by step solution

01

Understand the Base and Exponents

Identify the base and the exponent in the expression. In the expression \((0.5)^4\), the base is \(0.5\) and the exponent is \(4\). This means you must multiply \(0.5\) by itself a total of \(4\) times.
02

Multiplication of the Base

Multiply the base \(0.5\) by itself once: \(0.5 \times 0.5 = 0.25\). This is the first multiplication.
03

Continue Multiplying the Base

Multiply the result from Step 2 by \(0.5\) again: \(0.25 \times 0.5 = 0.125\). This is the second multiplication.
04

Final Multiplication

Multiply the result from Step 3 by \(0.5\) one last time: \(0.125 \times 0.5 = 0.0625\). This is the final multiplication.

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Key Concepts

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

Powers
When we talk about powers, we are dealing with expressions like \(a^n\), where \(a\) is the base and \(n\) is the exponent. These expressions are shorthand for repeated multiplication.
For instance, \((0.5)^4\) tells us to multiply \(0.5\) by itself \(4\) times. This is a way to simplify and speed up calculations involving the same number multiplied over and over again.
Powers turn multiple multiplications into a single expression with a base and an exponent. Using this, calculations become systematic and less prone to errors.
  • Visualizing it can help: Imagine stacking blocks; here, you add one more layer each time you multiply the base by itself.
  • Why use powers? They make math efficient, reduce room for error, and are foundational in more complex math concepts.
Base and Exponent
The base and exponent each play important roles. The base is the number you multiply, and the exponent indicates how many times you multiply that base. For \((0.5)^4\), \(0.5\) is the base, indicating the number being repeatedly multiplied, and \(4\) is the exponent, telling you to perform the multiplication \(4\) times.
A good way to interpret this is by considering the exponential "commands": the exponent directs the number of times you proceed with multiplication.
  • Key points: The base is the actor, doing the multiplication, while the exponent is like the script, telling it how many times to act.
  • An analogy: If the base is a recipe, the exponent tells you how many times to prepare that dish.
Multiplication
Multiplication is a fundamental math operation that can be visualized as repeated addition. When dealing with exponents, multiplication is at the core. As seen with the expression \((0.5)^4\), you multiply \(0.5 \,\times 0.5 \) then take that result and continue to multiply again until the exponent's requirement is satisfied.
In this situation, \(0.5 \times 0.5 = 0.25\); continue \(0.25 \times 0.5 = 0.125\); lastly, \(0.125 \times 0.5 = 0.0625\). This repeated multiplication helps achieve the final result the exponent calls for.
  • Process: Each step is building off the last, akin to climbing steps; each step is a multiplication.
  • Helpful tip: Keep track of results after each multiplication to ensure accuracy.

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Most popular questions from this chapter

Set up the following problems as you have been doing. Then use a calculator to perform the actual calculations. Round answers to two decimal places where appropriate. The Badlands National Monument in South Dakota was established in \(1929 .\) It covers \(243,302\) acres of land. What is the area in square miles?

Perform the indicated operations. $$10 \times 10 \times 10$$

Perform the indicated operations. $$8 \times 4$$

Perform the indicated operations. $$15+60$$

Perform the indicated operations. $$256 \div 640$$
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