Problem 33 - Elementary Algebra

Question

Multiply and reduce to lowest terms. $$ 12 \cdot 13 $$

Step-by-Step Solution

Answer

12 multiplied by 13 is 156, which is already in its simplest form.

Step 1: Multiply the Numbers

The first step in solving the expression is to multiply 12 by 13. Use long multiplication or mental math to calculate this product: \[12 \times 13 = 156\]

Step 2: Simplify the Result

Next, determine if 156 can be reduced to its lowest terms. Since this is a single number, we're actually simplifying the expression rather than the fraction. Check if 156 can be factored further: The factors of 156 are 1, 2, 3, 4, 6, 12, 13, 26, 39, 52, 78, 156. However, as it is already a single number and not a fraction, 156 is in its simplest form.

Key Concepts

Factors

Factors are numbers that can be multiplied together to get another number. Understanding factors is a fundamental skill in arithmetic, crucial for simplifying expressions or solving equations.

• For example, the factors of 12 are 1, 2, 3, 4, 6, and 12. These numbers can all be multiplied in various combinations to produce 12.
• Similarly, 13 is a prime number and has only two factors: 1 and 13.

In our example, after calculating the product of 12 and 13, we get 156. The factors of 156 include several numbers such as 1, 2, and 156. This is helpful to know when simplifying expressions, even though 156 in this exercise doesn't need further simplification.

Simplifying Expressions

Simplifying expressions involves reducing them to their simplest form. This could mean factoring out common numbers, canceling, or simply rewriting an expression to make it less complicated.

• In algebra, this often means collecting like terms or utilizing the distributive property when multiplying expressions that contain variables.
• However, in arithmetic, like our example, it often means finding common factors.

Even though 156 doesn't need to be reduced further, recognizing that it is already in its simplest form is part of simplifying. Simplifying can also involve ensuring all calculations are correct and that the final answer is as straightforward as possible.

Long Multiplication

Long multiplication is a method for multiplying two numbers—especially useful for more significant numbers or when mental math becomes challenging.

• Each digit of the first number is multiplied by each digit of the second number, starting from the rightmost digit and moving left.
• The products are then added together after being appropriately shifted based on their place value.

For instance, when you multiply 12 by 13 using long multiplication: 1. Multiply 3 by 12 to get 36. 2. Then, multiply 1 (from 13, which actually represents 10) by 12 to get 120. 3. Add these products: 36 + 120 = 156.
This step-by-step process shows the usefulness of long multiplication, making sure no number is missed and calculations are precise.