Problem 17 - Elementary and Intermediate Algebra

Question

Identify the base and the exponent in each expression. A. \(9 m^{12}\) B. \((9 m)^{12}\) C. \(-9 m^{12}\)

Step-by-Step Solution

Answer

A: Base is \(m\), exponent is 12. B: Base is \(9m\), exponent is 12. C: Base is \(m\), exponent is 12.

Step 1: Identify Base for Expression A

In the expression \(9 m^{12}\), the exponent, \(12\), is applied only to \(m\). Here, the base is \(m\).

Step 2: Identify Exponent for Expression A

Again referring to expression \(9 m^{12}\), the exponent is \(12\) associated with the base \(m\). Thus, the exponent is \(12\).

Step 3: Identify Base for Expression B

For expression \((9 m)^{12}\), the entire quantity \(9m\) is raised to the power of \(12\). Hence, the base is \(9m\).

Step 4: Identify Exponent for Expression B

The exponent in \((9 m)^{12}\) is \(12\), which is applied to the base \(9m\). Therefore, the exponent is \(12\).

Step 5: Identify Base for Expression C

In expression \(-9 m^{12}\), the exponent \(12\) only affects \(m\). Therefore, the base affected by the exponent is \(m\).

Step 6: Identify Exponent for Expression C

For expression \(-9 m^{12}\), the exponent is \(12\), used with the base \(m\), though the \(-9\) acts as a coefficient. Therefore, the exponent affecting \(m\) is \(12\).

Key Concepts

Expressions

In mathematics, expressions are like sentences of numbers and symbols put together. They represent values or operations. Each expression can consist of numbers, variables, and operations such as addition, subtraction, multiplication, and so on. Let's break down some essential components of an expression:

• Numerical expressions: These include just numbers and operations. For example, 5 + 3 or 6 × 2.
• Algebraic expressions: These contain variables, numbers, and operations. Examples include x + 3 or 4y - 2.

In our original exercise, we're looking at expressions like \(9 m^{12}\). This is an algebraic expression since it contains the variable \(m\). It's important to recognize that expressions do not contain an equals sign—they are simply combinations that represent a value or values.

Mathematical Notation

Mathematical notation is like the language of math. It helps us convey mathematical ideas clearly and concisely. Notation includes symbols, formulas, and signs that tell us how to understand and work with math problems and equations. Here are some key elements:

• Symbols: Such as \(+\), \(-\), \(\times\), \(\div\), \(^{\wedge}\) for powers, etc.
• Variables: Letters like \(x\), \(y\), or \(m\) that stand for unknown values or can represent quantities that vary.
• Exponents: A compact way to show that a number is multiplied by itself a certain number of times, for example, \(m^{12}\) which means \(m\) is multiplied by itself 12 times.

Using the right notation helps avoid misunderstandings and ensures consistency across various math problems. For instance, in \(9 m^{12}\), the exponent \(12\) is applied to the variable \(m\), telling us that \(m\) is raised to the twelfth power.

Algebraic Expressions

An algebraic expression combines variables, numbers, and operators to show a mathematical relationship. These expressions are foundational in algebra, allowing us to generalize mathematical principles and solve problems. Key parts of algebraic expressions include:

• Coefficients: Numbers that multiply variables. For instance, in \(-9 m\), \(-9\) is the coefficient of \(m\).
• Variables: Symbols like \(m\) representing numbers that can vary.
• Exponents: Numbers that show how many times to multiply the base by itself, like \(m^{12}\).
• Terms: Parts of the expression separated by plus or minus signs. In \(-9 m^{12}\), it's one term.

When we answer questions like identifying the base and exponent in an algebraic expression, it deepens our understanding of how these expressions work. For example, in \((9 m)^{12}\), the whole \(9m\) is the base, showing how these numbers and variables interact within the expression.