Problem 65 - Algebra 1: Concepts and Skills

Question

Write the expression as a single power of the base. $$ 2(2)^{4} $$

Step-by-Step Solution

Answer

The simplified expression as a single power of the base for \(2(2)^{4}\) is \(2^{5}\).

Step 1: Identify the base and powers

In our problem, we have \(2(2)^{4}\). Here, 2 is our base, and it is raised to the power of 4. The other 2 is also the same base, but it is not raised to any power. When a base does not have an explicit power, it is understood to have a power of 1.

Step 2: Apply the exponent multiplication rule

Following the rule that \(a^{m} \times a^{n} = a^{m+n}\), we can combine the two bases by adding their powers. This means combining \(2^{1}\) and \(2^{4}\) to get \(2^{1+4}\).

Step 3: Solve the exponent addition

Now that our expression is simplified to \(2^{1+4}\), we can simplify further by performing the addition operation in the exponent. \(1+4\) equals 5, so our final expression is \(2^{5}\).

Key Concepts

Base and Exponent

In mathematics, when we talk about exponents, we refer to two main components: the base and the exponent itself.
The base is the number that you're going to multiply by itself a certain number of times. The exponent tells us how many times to use the base in a multiplication.
For instance, in the expression \(2^4\), 2 is the base, and 4 is the exponent, signifying that we multiply 2 by itself 4 times, resulting in 16.
When a base has no visible exponent, like our base 2 in \(2(2)^4\), we assume that its exponent is 1. This is because any number raised to the power of 1 remains the same.

• Base: The foundational number
• Exponent: The number of times the base is multiplied by itself

Understanding this separation is crucial in solving problems, as it helps identify when and how to apply specific rules on simplifying or transforming expressions.

Exponent Rules

Exponent rules are a set of guidelines that help us understand how to handle expressions that include exponents. One fundamental rule is the multiplication rule, which is used when multiplying two powers of the same base.
This rule states that when you multiply like bases, you add their exponents together. Mathematically, this is expressed as \(a^m \times a^n = a^{m+n}\).
Let's consider its application in the exercise. We have the expression \(2(2)^4\). The hidden exponent of \(2\) is 1, making it \(2^1\). According to the rule, \(2^1 \times 2^4\) becomes \(2^{1+4}\), simplifying to \(2^5\).
In a nutshell:

• Multiplication Rule: Add the exponents if the bases are the same.
• Helps simplify expressions by reducing multiple power terms into a single one.

This technique simplifies complicated expressions, reducing potential errors in computation.

Simplifying Expressions

Simplifying expressions involving exponents revolves around reducing expressions to their simplest form. This reduces complexity and allows us to understand and solve problems more easily.
One critical aspect of simplification is clearly identifying like terms, especially those involving the same base.For the expression \(2(2)^4\), simplification involved:\(2^1 \times 2^4\).
We utilized the exponent multiplication rule to combine these into \(2^{1+4}\), which further simplifies to \(2^5\).

• Identify each component and its role.
• Apply exponent rules correctly.
• Perform arithmetic operations within the exponents.

By following these steps, you can ensure that your expression is in its simplest form, making calculations more straightforward and reducing the likelihood of mistakes.