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📘intermediate algebra review

key term - $a imes 10^n$

Citation:

Definition

$a imes 10^n$ is a mathematical expression that represents a number in scientific notation, where $a$ is a real number, and $n$ is an integer. This term is particularly relevant in the context of the properties of exponents and scientific notation, as it provides a concise way to express very large or very small numbers.

5 Must Know Facts For Your Next Test

  1. The term $a imes 10^n$ is used to represent numbers in scientific notation, which is a way of expressing very large or very small numbers in a compact form.
  2. The value of $a$ is the coefficient, which is a real number between 1 and 10, and the value of $n$ is the exponent, which is an integer that represents the power of 10.
  3. Scientific notation is useful for performing calculations with very large or very small numbers, as it simplifies the process of multiplying and dividing by powers of 10.
  4. The exponent $n$ in $a imes 10^n$ represents the number of places the decimal point needs to be moved to the right (for positive values of $n$) or to the left (for negative values of $n$) to express the number in standard form.
  5. The use of $a imes 10^n$ in scientific notation helps to maintain the appropriate number of significant figures, which is important for ensuring the accuracy of calculations and measurements.

Review Questions

  • Explain how the term $a imes 10^n$ is used to represent numbers in scientific notation.
    • The term $a imes 10^n$ is used to represent numbers in scientific notation, where $a$ is a real number between 1 and 10, and $n$ is an integer that represents the power of 10. This form of expression is useful for representing very large or very small numbers in a compact way. The exponent $n$ determines the number of places the decimal point needs to be moved to the right or left to express the number in standard form. For example, the number 6,230,000 can be written as $6.23 imes 10^6$, and the number 0.000045 can be written as $4.5 imes 10^{-5}$.
  • Describe how the use of $a imes 10^n$ in scientific notation helps to maintain the appropriate number of significant figures.
    • The use of $a imes 10^n$ in scientific notation helps to maintain the appropriate number of significant figures by ensuring that only the meaningful digits are included in the representation of a number. The coefficient $a$ is a real number between 1 and 10, which means that it will always have the same number of significant figures as the original number. The exponent $n$ then determines the placement of the decimal point, without affecting the number of significant figures. This is particularly important when performing calculations with very large or very small numbers, as it helps to ensure the accuracy of the results and prevent the loss of significant figures.
  • Analyze how the properties of exponents, such as $a^m imes a^n = a^{m+n}$, can be applied to manipulate and simplify expressions involving $a imes 10^n$.
    • The properties of exponents, such as $a^m imes a^n = a^{m+n}$, can be applied to manipulate and simplify expressions involving $a imes 10^n$ in scientific notation. For example, if you have the expression $3.5 imes 10^4 imes 2.1 imes 10^{-3}$, you can use the property $a^m imes a^n = a^{m+n}$ to simplify this expression. Specifically, you can rewrite the expression as $(3.5 imes 2.1) imes 10^{4 + (-3)} = 7.35 imes 10^1 = 73.5$. This demonstrates how the properties of exponents can be used to perform calculations with numbers in scientific notation more efficiently and accurately.