Problem 4 - Algebra 1: Concepts and Skills

Question

Use the product of powers property to write the expression as a single power of the base. \(2^{2} \cdot 2^{3}\)

Step-by-Step Solution

Answer

The expression simplifies to \(2^{5}\).

Step 1: Identifying Like Bases

The expression given has two bases which are the same (2). The corresponding exponents are 2 and 3.

Step 2: Applying product of powers property

The product of powers property states that when multiplying two powers with the same base, you can add the exponents. Hence, this will give us \(2^{(2+3)}\).

Step 3: Simplifying the Expression

Adding the exponents together gives \(2^{5}\).

Key Concepts

Exponents

Exponents are a way to express repeated multiplication of a number by itself. The number being multiplied is called the "base," and the number of times it is multiplied is called the "exponent." For example, in the expression \(2^3\), 2 is the base and 3 is the exponent. This essentially means multiplying 2 by itself 3 times: \(2 \times 2 \times 2 = 8\). Exponents make it easier to write and work with such large numbers.
Understanding exponents is crucial because it simplifies calculations especially when dealing with very large or very small numbers. They follow specific mathematical rules, such as the product of powers, that allow us to manipulate and simplify expressions efficiently. Learning these rules helps build a strong foundation for more complex algebraic concepts.

Same Base Multiplication

When multiplying expressions that have the same base, an important rule called the product of powers property is used. This rule is simple yet powerful: when you have two exponents with the same base, you can add the exponents together. For example, if you are given the expression \(2^2 \cdot 2^3\), you identify that the base is the same (which is 2).
The product of powers property can be outlined as follows:

• Ensure that the bases are identical, in this case, the base 2.
• Add the exponents of the terms being multiplied: \(2 + 3\).
• Write the expression as a single power: \(2^{2+3} = 2^5\).

This method greatly simplifies expressions without the need for long multiplication, making calculations both faster and more pleasant.

Simplifying Expressions

Simplifying expressions is an essential step in making complex mathematical problems more approachable. It involves using rules and properties to rewrite an expression in a simpler form, which can make it easier to work with or understand. In our case of using the product of powers property, simplifying the expression \(2^2 \cdot 2^3\) required us to apply the property and add the exponents.
After adding the exponents in our expression, which were 2 and 3, we get \(2^5\). This is the simplified form of the original expression. Simplification reduces the expression to a more manageable form and allows for further operations, whether it be more multiplication, division, or solving equations. Hence, understanding simplifying techniques is valuable not just in math class but in any problems that require problem-solving and logical analysis.