Problem 49 - College Algebra and Calculus: An Applied Approach

Question

Simplify the expression.\(5^{1 / 2} \cdot 5^{3 / 2}\)

Step-by-Step Solution

Answer

The simplified version of the expression \(5^{1 / 2} \cdot 5^{3 / 2}\) is 25.

Step 1: Identifying the law of exponents

Since we have an operation of multiplication between two exponents with the same base, the law of exponents that applies is 'When multiplying like bases, add the exponents'. So, the expression \(5^{1 / 2} \cdot 5^{3 / 2}\) can be written as \(5^{1/2 + 3/2}\).

Step 2: Adding the exponents

Add the exponents: \(1/2 + 3/2 = 2\). Replace in the initial expression with the result of the addition, so we have: \(5^{2}\).

Step 3: Simplifying the expression

Finally, we simplify the expression \(5^2\) which gives us 25.

Key Concepts

Exponent Multiplication

When you're dealing with exponents, especially in multiplication, there is a handy rule to make things easier. This rule is crucial when you have the same base in multiple terms. Let's break it down:

• When you multiply numbers that have exponents and share the same base, you can add their exponents together. Therefore, for an expression like this: \(a^m \cdot a^n\), the result is \(a^{m+n}\).

Keep in mind that this rule only works when the base, the big number that's affected by the exponent, is the same in both terms. This is why in the exercise above, where both numbers have a base of 5, we can simply add their exponents (\(1/2 + 3/2\)). This makes the multiplication much simpler by reducing it to a sum of exponents.

Simplifying Expressions

Simplifying expressions is like tidying up a room. You want to make everything as clean and simple as possible, which often means combining like terms or making calculations easier.

Sometimes, you'll start with complex problems that look hard to solve, but simplifying is all about making them manageable. When simplifying mathematical expressions, keep these points in mind:

• Look for similar or like terms. Terms with the same base and operation can often be combined.
• Always follow the rules of arithmetic, such as adding exponents when you're multiplying like bases.

In the example exercise, after applying the rule for multiplying exponents, it left us with the expression \(5^2\). This can be simplified further by calculating \(5^2\), which equals 25. Simplification is about making expressions as straightforward as possible, often leading to a single, clear answer.

Algebraic Expressions

Algebraic expressions are like mathematical sentences. They use numbers, variables, and operations to describe relations or quantities. Understanding how to manage these expressions is crucial for solving equations and real-world problems.

• Each part of an algebraic expression is called a term. A term can be a constant number, a variable (like \(x\)), or a combination of both.
• The operations in algebraic expressions can include addition, subtraction, multiplication, and division.

In the initial problem, we dealt with an expression set to be simplified. Though this example did not include variables, the principle is the same if it did. By understanding the structure and rules, like those involving exponents, algebraic expressions become easier to manipulate and solve. Algebra aims to make these expressions as intuitive as possible, helping you find solutions systematically.