Problem 21 - Algebra 1: Concepts and Skills

Question

Write the expression as a single power of the base. \((-2)^{3} \cdot(-2)^{3}\)

Step-by-Step Solution

Answer

The expression \((-2)^{3} \cdot(-2)^{3}\) can be rewritten as a single power of the base -2, i.e., \((-2)^{6}\).

Step 1: Identify the base and the exponents

The base in this expression is -2 and the exponents are 3. The expression appears twice, indicating multiplication.

Step 2: Apply the rule of exponents

The rule of exponents states that when you multiply powers with the same base, you add the exponents. In this case, you add the exponents 3 and 3, due to multiplication of \((-2)^{3}\) and \((-2)^{3}\). This gives us \((-2)^{3+3}\).

Step 3: Calculate the new exponent

Adding the exponents gives us a new exponent of 6. Hence, the expression can be written as \((-2)^{6}\).

Key Concepts

Understanding the Base in Exponentiation

In mathematics, exponentiation is a way of representing repeated multiplication of the same number by itself. The base is the number that gets multiplied. For example, in the expression \((-2)^{3}\), the base is \(-2\). This means \(-2\) is multiplied by itself 3 times:

• \(-2 \times -2 \times -2\)

The base can be any real number, including negative numbers, fractions, and decimals. The properties of the base directly influence the result the of expression, especially since multiplying negative numbers together can result in a positive or a negative product, depending on how many times it is multiplied. When working with the base, always remember to observe its sign and consider its role in the final result.

Applying the Product of Powers Rule

The product of powers rule is a fundamental principle of exponentiation. It states that when you multiply two powers that have the same base, you simply add the exponents together. This rule helps simplify expressions and make computations more manageable.For instance, in our original exercise, we have:

• \((-2)^{3} \cdot (-2)^{3}\)

Both terms have a base of \(-2\). Applying the product of powers rule, we add the exponents 3 and 3:

• \((-2)^{3+3}\)

This simplifies to \((-2)^{6}\). By using the product of powers rule, you can condense repeated multiplications into a simpler form, making it easier to handle complex equations. Additionally, ensure that the base remains consistent for this rule to apply; if the bases are different, the rule cannot be used.

Connecting Exponents through Addition

Addition of exponents is a straightforward process when dealing with multiplication of powers with the same base. This concept supports the idea that you can combine the multiplication of identical bases into a single power by adding their exponents.In practical terms, think of it this way: each exponent represents how many times the base is used as a factor. When combining these, you accumulate the total number of times the base is used. For example, adding the exponents 3 and 3 means the base \(-2\) is used as a factor 6 times:

• \((-2)^{3+3} = (-2)^{6}\)

This method streamlines calculations and minimizes the steps required to reach an answer. Grasping this concept is crucial for solving more complex exponentiation problems efficiently. Remember, the addition of exponents only works with multiplication of identical bases, reinforcing the importance of maintaining consistent bases throughout the operation.