Problem 7 - Algebra 1: Concepts and Skills

Question

Match the power with the words that describe it. $$ 4^{6} $$ A. four to the sixth power B. three to the seventh power C. seven to the third power D. six to the fourth power

Step-by-Step Solution

Answer

The correct match for the given numerical expression \(4^{6}\) is option A 'four to the sixth power'.

Step 1: Identify the base and power in the numerical expression

The numerical expression is \(4^{6}\). In this, the base is 4 and the exponent (also called power) is 6.

Step 2: Match the numerical expression with the right phrase

The phrase that correctly describes the numerical expression \(4^{6}\) is 'four to the sixth power'. Search for this phrase among the options given.

Step 3: Checking options for the correct match

Upon checking the options, the phrase 'four to the sixth power' corresponds to option A. All other options are incorrect because either the base or the power do not match.

Step 4: Confirming the answer

By following these steps correctly, it can be concluded that option A 'four to the sixth power' is the correct match to the numerical expression \(4^{6}\).

Key Concepts

Understanding Base and Power

When dealing with exponents, you will often hear the terms "base" and "power" or "exponent." But what do these mean? In simple terms, the **base** is the number that is being multiplied by itself. The **power** or **exponent** is the small number written at the top right of the base. This shows how many times the base is multiplied by itself.

Let's take a look at the expression \( 4^6 \). Here, **4** is the base, and **6** is the power. This means you multiply 4 by itself 6 times: \( 4 \times 4 \times 4 \times 4 \times 4 \times 4 \).

Remember that:

• The base determines the starting number you multiply.
• The power tells you how many times the base is repeated through multiplication.
• Exponents make it easier to write long repeated multiplication compactly.

Being familiar with these terms is crucial in understanding numerical expressions.

Exploring Numerical Expressions

A numerical expression represents a number using mathematical symbols like numbers and operators like addition, subtraction, multiplication, and division.

When exponents are involved, a numerical expression could be something like \( 4^6 \). This expression uses the base 4 and the exponent 6 to imply a repeated multiplication: \( 4 \times 4 \times 4 \times 4 \times 4 \times 4 \).

Here are a few key points about numerical expressions with exponents:

• They allow you to express very large numbers compactly.
• The base can be any integer or even a fraction.
• The exponent can be any positive integer, representing how many times the base is multiplied.
• Understanding numerical expressions requires recognizing the role of each element in the expression.

The elegance of numerical expressions lies in their simplicity and efficiency.

Identifying and Matching Expressions

Matching expressions is a skill of understanding and identifying the exact terms that describe a given mathematical phrase. Let's consider the example \( 4^6 \). In this form, your task is to identify both the base and the exponent to match the expression with its written description.

Here's how you can effectively match expressions:

• First, spot the base: In \( 4^6 \), the base is 4.
• Next, recognize the exponent: Here, it is 6.
• Translate the expression into words: 'Four to the sixth power.'
• Review your options and select the one that describes what you see in the problem.

Thus, for the expression \( 4^6 \), 'four to the sixth power' is the correct match. Practicing how to quickly identify these elements helps in tackling more complex problems involving exponents.