Question

In the expression \(a^{5}, a\) is called the_____of the expression.

Short answer

'a' is the base of the expression.

Step-by-step solution

Understanding Power Expression

In an expression of the form \(b^{n}\), 'b' is the base and 'n' is the exponent. The base is the number that is being multiplied and the exponent is the number of times the base is multiplied by itself.

Identifying the Base

Looking at the expression \(a^{5}\), it is clear that 'a' is the number that is being multiplied. Therefore, 'a' fits the definition of a base.

Concluding the Term

Consequently, in the expression \(a^{5}\), 'a' is the base.

Key concepts

Power Expression

A power expression is one of the fundamental concepts in algebra that represents repeated multiplication of the same factor. It's formatted as \( b^n \) where \( b \) is the base and \( n \) is the exponent.

For example, \( 3^4 \) signifies that 3 is multiplied by itself 4 times: \( 3 \times 3 \times 3 \times 3 \). In power expressions, the base can be any number, variable, or even a more complex algebraic term, as long as it is being raised to an integer power. The exponent dictates how many times the base appears as a factor in the multiplication.

Power expressions are not limited to positive integers as exponents. They can include zero, negative integers, fractions, and more, each affecting the base in different ways. For instance, an exponent of zero means the result is one (\( b^0 = 1 \) for any \( b \) not equal to zero), while negative exponents signify division (\( b^{-n} = 1/b^n \) for \( b \eq 0 \) and \( n > 0 \) ).

Exponent

The exponent in an algebraic expression refers to the superscript number that indicates how many times the base is used as a factor in multiplication. It is a powerful shorthand notation in algebra for representing exponential growth or decay, geometric series, and other mathematically significant patterns.

Understanding the role of exponents is key to mastering simplification of algebraic expressions, solving exponential equations, and analyzing functions. It's worth noting that an exponent is more than just a number; it's an operator that transforms the base. When an exponent is a positive whole number, it's quite straightforward. However, special cases of exponents include:

• Zero as an exponent: Any non-zero base to the power of zero is equal to one, reflecting the idea of an empty product.
• Negative integers as exponents: A negative exponent indicates division, such as \( b^{-n} = 1/b^n \).
• Rational exponents: Fractions as exponents represent roots, like \( b^{1/n} \) being the nth root of \( b \).

Exponents are critical in functions, allowing us to model real-world phenomena such as population growth, radioactive decay, and interest calculations with exponential functions.

Algebraic Terminology

Algebraic terminology is a language that is used to describe the components and operations within algebraic expressions and equations. It is essential to have a clear grasp of this terminology in order to communicate effectively about mathematical concepts and to solve algebraic problems accurately.

Some fundamental terms in algebra include:

• Variable: A symbol that represents an unknown quantity, typically a letter like \( x \), \( y \) or \( a \) in \( a^5 \).
• Coefficient: A number that is multiplied by a variable in an algebraic expression.
• Constant: A value that does not change and is not multiplied by a variable; in \( 7x + 4 \) the number 4 is a constant.
• Term: A single mathematical expression which could be a variable, a number, or a combination of both. In \( 5x^2 + 3x \), both \( 5x^2 \) and \( 3x \) are terms.
• Polynomial: An expression composed of multiple terms, such as \( x^2 + 3x + 4 \).

These terms are integral in identifying and manipulating different parts of an algebraic expression, whether it's for simplifying, factoring, expanding, or solving equations. Additionally, understanding the correct use of terminology helps in the transition to more advanced topics in algebra and other areas of mathematics.