Problem 6 - Elementary and Intermediate Algebra

Question

Fill in the blanks. The ____ of the term \(3 x^{7}\) is 7 because \(x\) appears as a factor 7 times: \(3 \cdot x \cdot x \cdot x \cdot x \cdot x \cdot x \cdot x\).

Step-by-Step Solution

Answer

The exponent of the term \(3x^7\) is 7.

Step 1: Identify the Parts of the Term

Begin by recognizing the term provided, which is \(3x^7\). This consists of a coefficient, \(3\), and the variable \(x\) raised to a power, 7.

Step 2: Understand the Concept of Exponent

Recall that in an expression of the form \(a^n\), the number \(a\) is the base, and \(n\) is the exponent. The exponent indicates how many times the base is used as a factor.

Step 3: Apply Exponent Concept to the Given Term

In the term \(3x^7\), the exponent is 7. This means that the base \(x\) appears as a factor 7 times, which can be expressed as \(x \cdot x \cdot x \cdot x \cdot x \cdot x \cdot x\).

Step 4: Fill in the Blank

Conclude that the blank should be filled with 'exponent' since the term 'exponent' is used to describe the number of times the variable \(x\) is used as a factor.

Key Concepts

Coefficient

In algebraic terms, a coefficient plays a crucial role, as it multiplies a term and indicates its contribution to the expression. In the term \(3x^7\), the coefficient is 3. It tells us how many times we are taking the term \(x^7\).
Coefficients can be positive, negative, or even a fraction. This affects the sign and magnitude of the entire term.

• A positive coefficient means the term keeps its sign.
• A negative coefficient flips the sign of the term.
• If the coefficient is zero, the entire term equals zero.

Understanding what a coefficient does is fundamental when simplifying and solving equations. Always remember to consider the coefficient's sign when performing any mathematical operations on an algebraic term.

Exponentiation

Exponentiation is a mathematical operation involving two numbers, the base and the exponent. In the term \(x^7\), the \(x\) is known as the base, and the number 7 is the exponent. Exponentiation tells us how many times to multiply the base. For \(x^7\), it means preserving \(x\) as a factor 7 times. This concept is essential in many areas of mathematics, including algebra, calculus, and beyond.

• Exponentiation significantly simplifies multiplication involving repeated factors.
• Exponents can be fractions, negative numbers, or even zero, expanding the concept further.

Grasping exponentiation provides a solid foundation for more advanced mathematical studies, aiding in the understanding of functions and equations.

Base and Exponent Relationship

The relationship between the base and the exponent is fundamental in mathematics. The base is the number being multiplied, and the exponent tells how many times the base is used as a multiplier. In terms like \(x^7\), \(x\) is multiplied by itself 7 times.
This relationship showcases the powerful nature of exponential growth and decay. It allows simple representation of large numbers and quantities that increase or decrease rapidly, such as populations or radioactive decay.

• When the exponent is positive, it shows growth.
• Negative exponents imply division or decay.
• An exponent of zero always results in a value of 1, as any non-zero base raised to the power of zero equals one.

Understanding the base and exponent relationship is key to solving many different types of problems in math and science.