Problem 106 - Intermediate Algebra

Question

3 because 8 is a perfect cube. $$ 80 $$ # Write each integer as a product of two integers such that one of the factors is a perfect cube. For example, write 24 as 8 # 3 because 8 is a perfect cube. $$ 80 $$

Step-by-Step Solution

Answer

80 can be expressed as the product 8 # 10 with 8 as a perfect cube.

Step 1: Find Prime Factorization of 80

First, find the prime factors of 80. Start dividing by the smallest prime numbers. 80 is divisible by 2: \[ 80 \div 2 = 40 \]Continue dividing by 2: \[ 40 \div 2 = 20 \]Again by 2: \[ 20 \div 2 = 10 \]And once more by 2: \[ 10 \div 2 = 5 \]Now, we have 5, which is a prime number itself. The prime factorization of 80 is: \[ 80 = 2^4 \times 5 \]

Step 2: Identify a Perfect Cube in the Factors

A perfect cube is something of the form \( a^3 \). From the factorization, we can see that \( 2^3 = 8 \) is a perfect cube. This supports the decomposition since \( 80 = 8 \cdot 10 \).

Step 3: Write 80 as a Product of Two Numbers

Using the identified perfect cube, write 80 as:\[ 80 = 8 \times 10 \]Here, 8 is the perfect cube (\( 2^3=8 \)), satisfying the condition.

Key Concepts

Prime Factorization

When you are asked to break down a number into its smaller building blocks, you need to use prime factorization. This process involves expressing the number as a product of prime numbers. Prime numbers are numbers that have no divisors other than 1 and themselves. For instance, 2, 3, and 5 are prime numbers.To find the prime factorization of 80, you begin by dividing it by the smallest prime number, which is 2. You keep dividing by 2 until you can no longer divide evenly. In the case of 80, it results in a sequence like this:

• 80 ÷ 2 = 40
• 40 ÷ 2 = 20
• 20 ÷ 2 = 10
• 10 ÷ 2 = 5

At this stage, 5 remains, which is a prime number itself. This means the prime factorization of 80 is represented as \( 80 = 2^4 \times 5 \). Each of these factors are the building blocks of the number 80.

Integers

Integers are the numbers we regularly use that can be positive, negative, or zero. They do not have fractions or decimals. Examples include -3, 0, and 15.In math problems, mentioning integers means you're dealing with whole numbers. When you're asked to express one integer as a product of other integers, like in the exercise with 80, it means you need to find two numbers that, when multiplied, give you the original number. For instance, writing 80 as a product of two integers, where one is a perfect cube, helps us understand how numbers can be decomposed into smaller, simpler numbers. In our case, that means showing 80 as \( 8 \times 10 \), where 8 is both a perfect cube and an integer.

Factors

Factors of a number are integers you can multiply together to get that number. For the number 80, its factors include 2, 4, 5, 8, 10, 16, 20, 40, and 80 itself. When looking for a pair of factors, one of which is a perfect cube, you find that 8 fits because it is \( 2^3 \). Understanding factors is critical when solving problems like this. It helps to break down a number into its simplest components and allows for multiple representations. Factors are fundamental in determining the composition of numbers, and this knowledge is essential in various mathematical applications such as simplifying expressions, solving equations, and more.