Problem 66 - Intermediate Algebra

Question

Evaluate each expression. See Example \(8 .\) $$ 4 \cdot 5^{3} $$

Step-by-Step Solution

Answer

The value of the expression is 500.

Step 1: Understand Exponents

The expression involves raising a number to a power, represented by the exponent. In this case, the base number is 5 and the exponent is 3, so we need to calculate \(5^3\).

Step 2: Compute the Exponentiation

Calculate \(5^3\) by multiplying the base, 5, by itself three times: \(5 \times 5 \times 5 = 25 \times 5 = 125\). So, \(5^3 = 125\).

Step 3: Multiply by the Coefficient

Now that we have evaluated \(5^3\) to be 125, we need to multiply this result by 4 as indicated in the expression: \(4 \times 125\).

Step 4: Perform the Multiplication

Complete the multiplication to find the answer: \(4 \times 125 = 500\).

Key Concepts

Multiplication

Multiplication is one of the basic operations in arithmetic, crucial for solving expressions like the one given. It's the process of adding a number to itself a certain number of times. For instance, multiplying 4 by 125 means you add 125 four times.

• Example: If you have 4 groups of 125 apples, you combine all those apples into one large group by multiplying:

This gives you a total of 500 apples. Multiplication helps in scaling numbers, making it possible to handle large quantities easily.
It saves time compared to adding the same number repeatedly. In mathematical expressions, multiplication is often represented by the symbol "\( \times \)" or sometimes by a dot "·". Tackle problems by taking each number in turn and combining them according to their given operation.

Base and Exponent

In mathematics, exponentiation is a valuable operation that involves two parts: the base and the exponent. The base is the number being multiplied, and the exponent tells you how many times to multiply the base by itself.
For example, in \( 5^3 \), 5 is the base and 3 is the exponent. This means you multiply 5 by itself three times.

• First, multiply the base by itself: \( 5 \times 5 = 25 \)
• Then, multiply by the base once more: \( 25 \times 5 = 125 \)

After these calculations, you find that \( 5^3 = 125 \). Understanding the difference between base and exponent is crucial as it changes how you approach calculations. Mastering exponentiation can simplify expressions and make complex calculations more manageable.

Order of Operations

To evaluate numerical expressions correctly, it is essential to follow the order of operations, often remembered by the acronym PEMDAS—Parentheses, Exponents, Multiplication and Division (left to right), Addition and Subtraction (left to right).
This rule ensures consistency in solving expressions.

• In the expression \( 4 \cdot 5^3 \), first calculate the exponent: \( 5^3 = 125 \).
• Then proceed with the multiplication: \( 4 \times 125 \).

By adhering to the order of operations, you ensure that each part of the expression is handled in the correct sequence.
The "Exponents" step comes before "Multiplication," meaning you should resolve any powers or square roots before multiplying literal numbers or terms. Following PEMDAS helps avoid mistakes and guarantees the accuracy of your answer.