Problem 1 - Algebra and Trigonometry

Question

Evaluate each expression. $$ 5^{2} \cdot 5 $$

Step-by-Step Solution

Answer

The expression evaluates to 125.

Step 1: Understand the Expression

You have the expression \(5^2 \cdot 5\). This is a multiplication of a power of five and another five.

Step 2: Simplify the Power

First, evaluate the power in the expression. \(5^2\) means \(5\) multiplied by itself, which is \(5 \cdot 5 = 25\).

Step 3: Multiply the Simplified Result by 5

Take the result from Step 2, which is 25, and multiply it by 5. This gives: \(25 \cdot 5 = 125\).

Key Concepts

Multiplication

Multiplication is a math operation where one number is added to itself a certain number of times. It's like finding the total number of items in several groups of the same size. In our example, we need to multiply numbers that include both a base number and a power. Multiplying powers can seem tricky at first, but the process becomes easy when we break it down into smaller parts.

• Identify the two numbers you wish to multiply together. Here, we work with the base number and a power.
• Multiply the base number raised to a power (which means the base is multiplied by itself a specific number of times) by the standalone number.
• This involves simple arithmetic steps – you can think of each multiplication process as repetitive addition."

Understanding multiplication this way helps make confusing-looking expressions like the one we're solving in this exercise easier and less intimidating.

Powers of Numbers

The power of a number signifies how many times to multiply the number by itself. It's a shortcut way to show repetitive multiplication. When we see something like \(5^2\), it tells us to multiply 5 by 5. It's important to note that the tiny number above the base number, called the exponent, informs us "how many times" the base number is used as a factor. Here's a breakdown:

• Base: In \(5^2\), our base is 5. This is the number being multiplied.
• Exponent: The 2 in \(5^2\) is the exponent. This tells us to multiply 5 two times (\(5 \cdot 5\)).
• Result: Solving \(5^2\) gives us 25, as multiplying 5 by itself once equals 25.

Recognizing how these elements work together allows us to handle powers of numbers easily in mathematical expressions.

Simplifying Expressions

Simplifying expressions might sound complicated, but it's merely a matter of making a complex problem easier to manage by performing certain operations. We take a mathematical expression and break it down to its simplest form. In the context of our example, this involves:

• Evaluating the power: This was done by calculating \(5^2\), which equals 25.
• Conducting the remaining operation: After evaluating the power, we multiply that result by the remaining number: \(25 \cdot 5\).
• Finding the simplest possible result: After solving \(25 \cdot 5\), we get 125.

Using this process ensures we can tackle math problems in a step-by-step method, minimizing errors and confusion. Simplifying helps make expressions clearer and results more understandable.