Problem 1 - College Algebra

Question

Evaluate each exponential expression in Exercises 1–22. $$ 5^{2} \cdot 2 $$

Step-by-Step Solution

Answer

50

Step 1: Computing Exponential Value

Begin by simplifying the exponential operation. In this case, \(5^{2}\) means multiplying 5 by itself. So we get \(5^{2} = 5 \cdot 5 = 25 \).

Step 2: Multiply with the Remaining Number

Now, multiply the result by the other number in the expression. This gives us \( 25 \cdot 2 = 50 \).

Key Concepts

Exponents

An exponent refers to the number of times a number, called the base, is multiplied by itself. It is a compact way of showing repeated multiplication. For example, in the expression \( 5^{2} \), the base is 5 and the exponent is 2.
This means you multiply 5 by itself once, equating to \( 5 \times 5 = 25 \).
Exponents make it easier to write and calculate large numbers. Instead of writing five hundred 2s multiplied together, you simply write \( 2^{500} \). Here's a quick recap of exponent terminology:

• Base: The number that is multiplied.
• Exponent: Indicates how many times the base is used as a factor.

Understand that an exponent does not simply multiply the base by the exponent itself but uses it as a guide for how many times to multiply the base by itself.

Multiplication

Multiplication is one of the basic operations of arithmetic and is important for handling exponential expressions. In multiplication, you combine groups of numbers into a sum. In the case of our exercise, \( 25 \cdot 2 \), you're combining two groups of 25.
To visualize, imagine two stacks of 25 items each. Combining them gives you a total of 50 items.
The properties of multiplication you should remember are:

• Commutative Property: Means \( a \cdot b = b \cdot a \). Order doesn't matter.
• Associative Property: Changing grouping in multiplication doesn't change the result, \( (a \cdot b) \cdot c = a \cdot (b \cdot c) \).
• Distributive Property: \( a(b + c) = ab + ac \). This connects multiplication over addition.

Multiplication simplifies repeated addition, hence a cornerstone in calculus and algebra.

Order of Operations

When solving mathematical expressions that involve more than one operation, like addition, subtraction, multiplication, and division, following the correct order of operations is critical.
A simple way to remember the order is through the acronym PEMDAS:

• P: Parentheses first
• E: Exponents (i.e., powers and roots, etc.)
• MD: Multiplication and Division (left to right)
• AS: Addition and Subtraction (left to right)

In the given exercise, even though \( 5^{2} \cdot 2 \) has both exponents and multiplication, you solve the exponent part first due to the order of operations. Calculate \( 5^{2} \) before proceeding with multiplying by 2. Ignoring PEMDAS can lead to incorrect solutions, so always perform calculations in the right sequence to ensure accuracy.