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Chapter 5: Problem 86

Write each number in decimal notation without the use of exponents. $$8.6 \times 10^{-1}$$

Short Answer

Expert verified
The decimal notation of \(8.6 \times 10^{-1}\) is 0.86.

Step by step solution

01

Identify the Base and the Exponent

In this expression \(8.6 \times 10^{-1}\), 8.6 is the base or coefficient, and -1 is the exponent.
02

Understand the Meaning of the Negative Exponent

A negative exponent means that the number is less than 1. In this case, we have an exponent of -1, indicating that we move the decimal one place to the left to create a smaller number.
03

Move the Decimal Point

By moving the decimal one place to the left, the number 8.6 becomes 0.86.

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Key Concepts

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

Exponents
Exponents are a fundamental concept in mathematics that express how many times a number, known as the base, is multiplied by itself. For example, in the expression \(10^2\), 10 is the base and 2 is the exponent, representing the calculation \(10 \times 10 = 100\). Exponents simplify expressions, particularly in large numbers.
They also indicate repeated operations. Understanding exponents helps in performing calculations efficiently and is foundational for progressing in mathematics.
Key points to remember include:
  • The base is the number being multiplied.
  • The exponent denotes the number of times the base multiplies itself.
  • Exponents of 1 mean the number remains the same (e.g., \(10^1 = 10\)).
  • Exponents of 0 denote a value of 1 regardless of the base (e.g., \(10^0 = 1\)).
Recognizing these patterns aids in understanding more complex mathematical expressions.
Negative Exponents
Negative exponents represent numbers less than one, and they denote the reciprocal of the base raised to the equivalent positive exponent. For example, \(10^{-1} = \frac{1}{10^1} = 0.1\).
This means you are expressing the base's reciprocal and reflects a change in direction on the number line compared to positive exponents.
Let's explore how negative exponents work:
  • The expression \(10^{-1}\) translates to moving the decimal one place to the left, resulting in 0.1.
  • Similarly, \(10^{-2}\) becomes 0.01, moving the decimal two places to the left because it represents \(\frac{1}{10^2}\).
  • Negative exponents often occur in scientific notation, providing a concise way to express small numbers.
Handling negative exponents is crucial for converting between scientific and decimal notation.
Mathematics Education
Mathematics education is about more than just solving equations; it aims to build a strong conceptual understanding of mathematical principles such as exponents. Students often encounter challenges when concepts are not clearly linked to practical examples or daily applications. Here are strategies to enhance learning and comprehension:
  • Use real-world examples, such as moving decimal points when dealing with negative exponents, to connect abstract concepts to tangible experiences.
  • Encourage visualization through graphs or number lines to aid in understanding the impact of exponents on numbers.
  • Implement interactive activities that involve manipulating exponents and bases to see immediate results.
Effective mathematics education empowers students to tackle complex problems and builds confidence in their mathematical abilities. Ensuring they understand the 'why' and 'how' of concepts like exponents lays a foundation for advanced mathematics courses and real-life problem-solving.

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Most popular questions from this chapter

Find each of the products in parts (a)-(c). a. \((x-1)(x+1)\) b. \((x-1)\left(x^{2}+x+1\right)\) c. \((x-1)\left(x^{3}+x^{2}+x+1\right)\) d. Using the pattern found in parts (a)-(c), find $(x-1)\left(x^{4}+x^{3}+x^{2}+x+1\right) without actually multiplying.

Express \(\frac{7}{8}\) as a decimal. (Section \(1.3,\) Example 4 )

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. $$4^{-2}<4^{-3}$$

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. $$5^{2} \cdot 5^{-2}>2^{5} \cdot 2^{-5}$$

Determine whether each statement "makes sense" or "does not make sense" and explain your reasoning. If \(5^{-2}\) is raised to the third power, the result is a number between 0 and 1
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