Problem 58 - Algebra 1: Concepts and Skills

Question

Find the number with the given prime factorization. (Skills Review p. 761) $$ 5^{3} \cdot 7 \cdot 11 $$

Step-by-Step Solution

Answer

The number with the given prime factorization is 9625.

Step 1: Understanding the exponent notation

The notation \(5^{3}\) means 5 is multiplied by itself 3 times. So, calculate this first.

Step 2: Calculating the exponential expression

Calculate \(5^{3}\) = 5 × 5 × 5 = 125.

Step 3: Multiplying the results

Now, multiply this result with the other numbers given in the factorization. Hence, calculate 125 × 7 × 11.

Step 4: Calculating the final result

The multiplication 125 × 7 × 11 = 9625. Hence, the number with the given prime factorization is 9625.

Key Concepts

Exponent Notation

Exponent notation is a way of representing repeated multiplication of the same factor. When we write numbers in the exponent form, we are making the mathematical expressions more compact and easier to read. For example, in the expression (5^{3}), the number 5 is called the base and 3 is the exponent, indicating that the base is to be multiplied by itself as many times as the value of the exponent, which is three times in this case. Thus, (5^{3}) simplifies to 5 multiplied by 5, and then by 5 again, which equals 125.

Exponential Expression

An exponential expression consists of a base raised to a certain power, known as the exponent. The base is the number that is being multiplied, and the exponent tells us how many times the base multiplies itself. In the context of our problem, (5^{3}) is an exponential expression where 5 is the base and 3 is the exponent. Calculating an exponential expression like (5^{3}) is straightforward: take the base (5) and multiply it by itself the number of times indicated by the exponent (3).

It's essential to evaluate exponential expressions before carrying out any further operations, such as multiplication or division, to follow the correct order of mathematical operations.

Multiplying Prime Factors

To find a number's prime factorization means breaking the number down into its prime factors—numbers that are only divisible by 1 and themselves. When we're given the prime factors of a number, as in our exercise, we multiply these factors together to find the original number. In our case, we have three prime factors: 5, which appears three times, 7, and 11. We multiply these primes in several steps: first, resolve the exponent, then multiply the result by the remaining prime factors.

So, in our exercise, after calculating (5^{3} = 125), we then multiply 125 by 7 and 11 to find the original number. This process is straightforward when dealing with a small number of prime factors but become increasingly complex with larger prime factorizations. It's beneficial to use exponentiation to simplify the expression and reducing potential calculation errors.