Problem 3 - Prealgebra

Question

Find each product or quotient. Express using exponents. $$6 \cdot 6^{6}$$

Step-by-Step Solution

Answer

The product is \( 6^7 \).

Step 1: Understand Exponent Properties

When multiplying expressions with the same base, you add the exponents. In this case, our base is 6. Applying this property, we have: \[ a^m \cdot a^n = a^{m+n} \].

Step 2: Apply the Property to the Given Problem

The given problem is \( 6 \cdot 6^6 \). We can rewrite 6 as \( 6^1 \) because any number raised to the power of 1 is itself. Now, we have \( 6^1 \cdot 6^6 \).

Step 3: Add the Exponents

Using the property from Step 1, add the exponents: \[ 6^1 \cdot 6^6 = 6^{1+6} = 6^7 \].

Step 4: Express the Product

The product, expressed using exponents, is \( 6^7 \).

Key Concepts

Product

In mathematics, the term "product" refers to the result obtained when two or more numbers are multiplied together. Consider the simple multiplication operation such as multiplying 3 by 4. The product of 3 and 4 is 12. Products help us consolidate repeated addition into a single operation, making calculations efficient.

• Product arises in multiplication.
• It simplifies repetitive addition.
• Expressed in mathematical notation often using the multiplication sign \( \cdot \).

In the context of the exercise, we are asked to find the product of the numbers in an exponential format. For instance, taking the expression \( 6 \cdot 6^6 \), the product here is found by correctly applying the rules of exponents.

Multiplying Exponents

Multiplying expressions that contain exponents follows specific mathematical rules. When you have the same base numbers and you're multiplying them, the exponents of those bases are simply added together. This property makes calculations involving powers straightforward and efficient.
For example, the law \( a^m \cdot a^n = a^{m+n} \) shows how exponents are computed during multiplication. For simple numbers, like \( 6^1 \cdot 6^6 \), you keep the base the same and add the exponents: 1 and 6. This results in \( 6^7 \).

• Applicable only when bases are identical.
• Simplifies large number multiplications.
• Reduces complexity by combining into one exponent term.

This rule turns what could be lengthy calculations into manageable expressions by treating the exponents as parts of a larger equation.

Base Number

The base number in an exponential expression is the number that is repeatedly multiplied by itself according to the exponent value. In the term \( 6^6 \), the number 6 is the base, and it is what you would use as a starting point for that multiplication.
The base plays a critical role because it defines the main number being affected by the power. Each base, when paired with an exponent, gives rise to different values and solutions.

• Identifies the numeral being multiplied.
• Holds key role in determining the magnitude of power.
• Assists in understanding relationships in algebraic expressions.

Comprehending the base leads to a better grasp of how exponents alter a number. Without this fundamental part of exponents, interpreting expressions like \( 6 \cdot 6^6 \) would be challenging, especially when the importance of maintaining base consistency during multiplication is paramount.