7.1 Power Notation and Meaning

Power Notation and Meaning simplifies repeated multiplication using exponents, making mathematical expressions clearer and more efficient.

Power Notation and Meaning describes the compact notation used to represent repeated multiplication of a single number by itself, including the roles of the base and exponent, the standard vocabulary for common powers, and how a power is evaluated to a numerical value.

The Components of a Power

Base and Exponent Roles

A power consists of two parts: the base, which is the number being multiplied, and the exponent, which indicates how many times the base is used as a factor. The exponent is written as a small raised number to the upper right of the base.

b^{n}

Power as Repeated Multiplication

A power expands into a product of the base multiplied by itself as many times as the exponent specifies. The exponent counts the number of factors, not the number of multiplication operations, so an exponent of 3 means three copies of the base multiplied together, using two multiplication signs.

5^{3} = 5 \times 5 \times 5

Power Reading and Interpretation

A power is read aloud according to its exponent: an exponent of 2 is read as "squared," an exponent of 3 is read as "cubed," and any higher exponent is read as "to the power of" followed by the exponent's value, or as "to the nth" for a general exponent n.

Positive Integer Powers

Positive Integer Exponents

When the exponent is a positive whole number, the power represents a finite, well-defined product of that many copies of the base, and this is the most basic and intuitive case of exponentiation from which other exponent rules are later extended.

First Power

A base raised to the first power equals the base itself, since a single copy of the base with no additional multiplication is simply the base's own value.

b^{1} = b

Squared Number Notation

A number raised to the second power is called the square of that number, written with an exponent of 2. Squaring a number corresponds geometrically to computing the area of a square whose side length equals that number.

7^{2} = 7 \times 7 = 49

Cubed Number Notation

A number raised to the third power is called the cube of that number, written with an exponent of 3. Cubing a number corresponds geometrically to computing the volume of a cube whose side length equals that number.

4^{3} = 4 \times 4 \times 4 = 64

Higher Positive Integer Powers

Exponents beyond 3 continue the same pattern of repeated multiplication and are named by their numeric value, such as "to the fourth power" or "to the fifth power," without a distinct geometric shorthand comparable to squaring or cubing.

Evaluating and Distinguishing Powers

Power Value Evaluation

Evaluating a power means carrying out the repeated multiplication to arrive at a single numerical result. This is done by multiplying the base by itself the number of times indicated by the exponent, combining the factors from left to right until one value remains.

2^{5} = 2 \times 2 \times 2 \times 2 \times 2 = 32

Power and Multiplication Coefficient Distinction

A power must be distinguished from a coefficient placed in front of a variable or number through ordinary multiplication. The expression 3 times x means three copies of x added together, while x raised to the third power means three copies of x multiplied together; the two operations produce different results and must not be interchanged.

3 x \neq x^{3}