Problem 64 - Algebra 1: Concepts and Skills

Question

Write the expression as a single power of the base. $$ 5 \cdot 5^{2} \cdot 5^{3} $$

Step-by-Step Solution

Answer

The expression \(5 \cdot 5^{2} \cdot 5^{3}\) as a single power of the base is \(5^{6}\).

Step 1: Identify the base and the exponents

In the given expression \(5 \cdot 5^{2} \cdot 5^{3}\), the base is number 5, and the exponents are 1 (implied from 5), 2, and 3 respectively.

Step 2: Apply the rule of product of powers

Using the product of powers rule, the expression becomes \(5^{1+2+3}\). Here, the exponents are added because the bases are the same.

Step 3: Simplify the new exponent

The exponent now becomes \(5^{1+2+3} = 5^{6}\).

Key Concepts

Understanding Exponents

An exponent is a fundamental concept in mathematics, representing the number of times a base is used as a factor in a multiplication. For example, in the expression \( 5^{3} \), the base is 5 and the exponent is 3. This means \(5\) is multiplied by itself three times: \(5 \times 5 \times 5\).

Exponents can help us express repeated multiplication concisely, making our calculations quicker and more manageable. They're not only a math tool but also present in sciences, such as physics and engineering, where large or small quantities are often expressed using powers of ten.

• The base tells us what number we're multiplying.
• The exponent shows how many times the base is used as a factor.
• An exponent of 1 means the base itself, while an exponent of 0 gives us 1 (for any non-zero base).

Product of Powers Rule

The product of powers rule is a handy tool when dealing with multiplication of expressions with the same base. This rule states that when multiplying like bases, you add the exponents together.

If you have a base \(a\), the product of powers rule looks like this: \(a^{m} \cdot a^{n} = a^{m+n}\). This property simplifies calculations by reducing lengthy multiplication to simple addition of exponents.

In our original problem, \(5 \cdot 5^{2} \cdot 5^{3}\), each \(5\) has the same base. Hence, by applying the product of powers rule, we can simplify it to \(5^{1+2+3}\), resulting in a single expression: \(5^{6}\).

• The rule applies only to expressions with the same base.
• Adding exponents saves you from multiplying the base repeatedly.

Simplifying Expressions

Simplifying expressions is about making them more compact and easier to work with, particularly when preparing them for further calculations. In mathematics, simplifying often means rewriting an expression in a more concise form without changing its value.

Using the product of powers rule, as shown in the solution, helps reduce multiple terms to a single term. For instance, our expression \(5 \cdot 5^{2} \cdot 5^{3}\) is simplified to \(5^{6}\), easing subsequent mathematical operations.

• Simplification often involves reducing multiplications and additions.
• It makes expressions easier to read and work with, especially in larger equations.
• You aim to keep the base the same and manipulate the exponents effectively.

By simplifying expressions, you improve computational efficiency and clarity, whether you're handling simple calculations or complex equations.