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The introduction covers the importance of exponents in simplifying large numbers, such as the mass of the Earth, and illustrates the concept of negative exponents with examples. It discusses how negative exponents represent fractions, aiding in calculations involving numbers less than one.
Exponents are mathematical notations that allow us to express large numbers efficiently. For instance, the mass of the Earth is expressed using exponents as 5.97 × 10²⁴ kg instead of writing it out in full. In this section, we will explore the meaning of exponents, focusing on positive and negative integers.
We begin by reviewing how a number raised to a power represents repeated multiplication (e.g., 2⁵ = 2 × 2 × 2 × 2 × 2). We then delve into negative exponents, illustrating that an exponent like 10⁻² can be rewritten as 1/(10²), or 1/100, which equals 0.01. Through a series of patterns and examples, we learn how to compute values involving negative exponents and express numbers in expanded form. Understanding exponents is crucial in higher mathematics and helps simplify complex numerical situations.
Exponents simplify the expression of large and small numbers.
Negative exponents represent the reciprocal of the base raised to the absolute value of the exponent.
The expanded form expresses numbers as the sum of their parts using powers of ten.
Exponents are neat, they can’t be beat, powers of ten are simple and sweet.
Imagine a tiny ant representing a negative exponent; when the ant follows the pattern, it climbs down into the denominator.
Remember, Negative means Divide: Just think of it as taking a number down to the basement, below one.
Mass of the Earth: 5,970,000,000,000,000,000,000,000 kg = 5.97 × 10²⁴ kg.
0.000007 m = 7 × 10⁻⁶ m; this translates into a very small number using negative exponents.
Converting 1425 to expanded form: 1 × 10³ + 4 × 10² + 2 × 10¹ + 5 × 10⁰.
Term: Exponent
Definition: A mathematical notation indicating the number of times a number (the base) is multiplied by itself.
A mathematical notation indicating the number of times a number (the base) is multiplied by itself.
Term: Negative Exponent
Definition: An exponent indicating the reciprocal of the base raised to the absolute value of the exponent.
An exponent indicating the reciprocal of the base raised to the absolute value of the exponent.
Term: Expanded Form
Definition: A way of expressing a number as the sum of its individual digits multiplied by their place values.
A way of expressing a number as the sum of its individual digits multiplied by their place values.
Term: Base
Definition: The number that is being raised to a power in an exponential expression.
The number that is being raised to a power in an exponential expression.
Term: Multiplicative Inverse
Definition: The reciprocal of a number; for example, the multiplicative inverse of 2 is 1/2.
The reciprocal of a number; for example, the multiplicative inverse of 2 is 1/2.