Problem 16 - Elementary and Intermediate Algebra

Question

Identify the base and the exponent in each expression. A. \(-\frac{1}{3} x^{6}\) B. \(\left(-\frac{1}{3} x\right)^{6}\) C. \(-\left(-\frac{1}{3} x\right)^{6}\)

Step-by-Step Solution

Answer

A: Base = \( x \), Exponent = 6; B & C: Base = \(-\frac{1}{3} x\), Exponent = 6.

Step 1: Understanding Base and Exponent

In the expression given as \( a^b \), \( a \) is called the base and \( b \) is called the exponent. The base is the number or variable being multiplied by itself and the exponent tells us how many times the base is used as a factor.

Step 2: Analyzing Expression A

For the expression \(-\frac{1}{3} x^{6}\), identify the base and the exponent. The base is \( x \) because it is the number being raised to a power. The exponent is \( 6 \) because it indicates the number of times the base \( x \) is multiplied by itself.

Step 3: Analyzing Expression B

Consider expression \( \left(-\frac{1}{3} x\right)^{6} \). Here, the entire term \(-\frac{1}{3} x\) is enclosed in parentheses and raised to the power of 6. So, \(-\frac{1}{3} x\) is the base, and the exponent is \( 6 \). This means the whole expression is being multiplied by itself six times.

Step 4: Analyzing Expression C

For expression \(-\left(-\frac{1}{3} x\right)^{6}\), notice that the minus sign outside is not part of the base or the exponent. The expression within the parentheses \(-\frac{1}{3} x\) is the base of the power, and \( 6 \) is the exponent. The minus sign outside negates the entire result of the expression inside the parentheses raised to the power.

Key Concepts

Understanding Base and Exponent

When working with exponents, two primary elements are involved: the base and the exponent. These are vital components of any expression involving powers, such as in a simple expression like \( a^b \). Here, \( a \) represents the base, and \( b \) denotes the exponent.
Understanding these parts is crucial because:

• The base is the number or variable that is being multiplied by itself.
• The exponent shows how many times the base is used as a factor.

For example, in the expression \( 3^4 \), **3** is the base, and it means 3 is multiplied by itself four times: \( 3 \times 3 \times 3 \times 3 \). This results in a value of 81. Being able to identify these components in an equation helps in simplifying complex mathematical expressions and problems.

Algebraic Expressions and Their Components

Algebraic expressions contain variables, numbers, and operations. Expressions with exponents are a common part of algebra, where identifying the base and exponent simplifies your calculations. For example, let's look at expression \(-\frac{1}{3} x^6\):
- Here, **x** is the base because it's the variable being raised to a power.
- The number **6** is the exponent, showing that \( x \) is used as a factor six times.In expressions like \( \left(-\frac{1}{3} x\right)^6 \), the situation is slightly different:

• The entire term inside the parentheses, \(-\frac{1}{3} x\), is considered the base.
• This whole expression is raised to the power 6, meaning \(-\frac{1}{3} x\) is multiplied by itself six times.

Finally, consider \(-\left(-\frac{1}{3} x\right)^6\):
- Notice that the negative sign outside does not affect the base directly.
- The base remains \(-\frac{1}{3} x\), which is then raised to the exponent of 6. The negative sign outside negates the entire value of the expression calculated. Understanding these elements is essential for mastering any algebraic manipulation involving exponents.

Importance of Step-by-Step Solutions

Learning mathematics can be much easier when problems are broken down into understandable steps. Step-by-step solutions are beneficial because they:

• Provide clarity by focusing on each component of the problem one at a time.
• Allow students to see the reasoning behind each calculation.
• Help in identifying mistakes by following a logical path.

For instance, in solving the expressions we've looked at, breaking the solution into steps enables you to: - Identify what forms the base and the exponent separately. - Understand how the parentheses influence which part of the expression is considered the base. - Comprehend the impact of an external negative sign on the final result. By carefully analyzing each part of an expression, students can build their confidence in handling more complex algebraic concepts. Step-by-step approaches transform challenging equations into simple, manageable parts.