Problem 6 - Algebra 1

Question

Use the product of powers property to simplify the expression. $$ 2^{2} \cdot 2^{3} $$

Step-by-Step Solution

Answer

The simplified form of \( 2^{2} \cdot 2^{3}\) using the product of powers property is \( 2^{5}\).

Step 1: Identifying the Bases and Exponents

The given expression is \(2^{2} \cdot 2^{3}\). Here, the base is 2 and the exponents are 2 and 3.

Step 2: Applying the Product of Powers property

According to the Product of Powers property, when multiplying bases of the same value, you can add the exponents together. Here the base is the same (2) for both terms, so add the exponents: \(2^{(2+3)}\).

Step 3: Solving the Expression

Now solve the expression: \(2^{(2+3)} = 2^{5}\).

Key Concepts

Exponents

An exponent, often referred to as the power, is a number that indicates how many times the base, which is another number, is multiplied by itself. For instance, in the expression \(2^3\), the number 3 is the exponent, and it tells us that we should multiply the base, which is 2, by itself three times (2 x 2 x 2). Exponents are a shorthand way to express repeated multiplication of the same number and are an integral part of algebraic operations.

Base

In the realm of exponents, the base is the number that is being raised to a power. In other words, it's the number that is getting multiplied by itself a certain number of times as indicated by the exponent. Sticking with our earlier example, within \(2^3\), the base is 2. The base is essentially the 'subject' of the exponential expression, and it is crucial in determining the value of the expression. A solid understanding of the base is necessary when simplifying expressions that involve exponents.

Simplifying Expressions

Simplifying expressions is a critical skill in mathematics that involves rewriting equations or expressions in their most basic form. Simplification might involve several properties including combining like terms, using the distributive property, or, as in our example, applying the product of powers property. The ultimate goal is to make an expression as simple and as easily understandable as possible without changing its value. In simplifying \(2^{2} \cdot 2^{3}\), we leveraged the product of powers property to combine the exponents, resulting in a more streamlined expression: \(2^5\).

Exponential Notation

Exponential notation is a way of representing numbers that are too large or too small to be conveniently written in standard decimal form. It uses a base and an exponent to succinctly express the number. In exponential notation, a number is written as the product of a base raised to an exponent, like so: \(b^n\), where \(b\) is the base and \(n\) is the exponent. This form of notation is incredibly valuable across all branches of mathematics and science since it simplifies the handling and calculation of large or small numbers.