📘intermediate algebra review

key term - $10^n$

Definition

$10^n$ is a mathematical expression that represents a power of 10, where $n$ is an exponent. This term is particularly relevant in the context of the properties of exponents and scientific notation, as it allows for the concise representation of very large or very small numbers.

5 Must Know Facts For Your Next Test

  1. $10^n$ can be used to represent very large or very small numbers in a compact form.
  2. The exponent $n$ determines the magnitude of the number, with positive exponents representing large numbers and negative exponents representing small numbers.
  3. When the exponent $n$ is 0, $10^0$ is equal to 1.
  4. The properties of exponents, such as $10^a \times 10^b = 10^{a+b}$, can be used to simplify expressions involving $10^n$.
  5. Scientific notation utilizes $10^n$ to express numbers in a standardized format, making it easier to compare and perform calculations with very large or very small numbers.

Review Questions

  • Explain how the value of the exponent $n$ in $10^n$ affects the magnitude of the number.
    • The value of the exponent $n$ in $10^n$ determines the magnitude of the number. When $n$ is positive, $10^n$ represents a large number, with the magnitude increasing as the exponent gets larger. For example, $10^3 = 1,000$ and $10^6 = 1,000,000$. Conversely, when $n$ is negative, $10^n$ represents a small number, with the magnitude decreasing as the exponent becomes more negative. For instance, $10^{-3} = 0.001$ and $10^{-6} = 0.000001$.
  • Describe how the properties of exponents can be used to simplify expressions involving $10^n$.
    • The properties of exponents can be used to simplify expressions involving $10^n$. For example, the property $10^a \times 10^b = 10^{a+b}$ can be used to combine multiple powers of 10 into a single power. Additionally, the property $\dfrac{10^a}{10^b} = 10^{a-b}$ can be used to divide powers of 10. These properties allow for the efficient manipulation and representation of very large or very small numbers using $10^n$.
  • Explain how $10^n$ is used in the context of scientific notation and discuss the benefits of this representation.
    • Scientific notation utilizes $10^n$ to express numbers in a standardized format. In scientific notation, a number is written as a product of a number between 1 and 10, and a power of 10. For example, the number 12,345 can be written as $1.2345 \times 10^4$ in scientific notation. This representation offers several benefits, such as the ability to easily compare and perform calculations with very large or very small numbers, as well as the compact and efficient expression of such quantities. The use of $10^n$ in scientific notation is a crucial tool for working with and understanding numbers of vastly different magnitudes.

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