Question

Use mental math to find the product. $$-26 \cdot 34$$

Short answer

The product of -26 and 34 is -884.

Step-by-step solution

Identify the Values

To start off, we identify the values that need to be multiplied. Our values are -26 and 34.

Simplify the Multiplication

Instead of multiplying -26 and 34 directly, we can break it down. We can view -26 as -20 and -6. So now we will calculate -20 times 34 and -6 times 34.

Multiply -20 by 34

Multiplying -20 by 34 gives us -680.

Multiply -6 by 34

Multiplying -6 by 34 produces -204.

Combine for Final Value

Adding the results from step 3 and step 4 together, -680 + -204, gives us -884. This is our final value.

Key concepts

Negative Number Multiplication

Understanding how to multiply negative numbers is a fundamental mathematical skill. At first glance, it might seem a bit intimidating, but with some basic rules, it gets much simpler. When multiplying two numbers where one or both are negative, you must remember the following vital rule: a negative times a positive gives a negative, while a negative times a negative gives a positive.
In the exercise \( -26 \times 34 \) we have a negative times a positive, which means our result will definitely be negative. So when you approach problems involving negative numbers, focus on the magnitude of numbers first, multiply them as if they were positive, and then apply the negative sign according to the rule.

Break Down Multiplication

Breaking down multiplication into more manageable chunks can make mental math much easier. This technique involves splitting one or both of the numbers into parts, preferably tens, hundreds, or other numbers that are easy to multiply mentally.
For instance, in the exercise where we multiply \( -26 \times 34 \) by breaking down -26 into \( -20 \) and \( -6 \) we're creating two calculations that are simpler to handle. After calculating each part's product, you add them together to get the final answer. This is particularly helpful for large numbers or in situations where you're trying to calculate quickly without paper and pen. The convenience of breaking down numbers is seen in its ability to turn a daunting multiplication task into a series of smaller, more feasible calculations.

Simplify Multiplication

The ultimate goal of simplifying multiplication is to reduce complexity and make calculation easier. There are several methods to simplify multiplication including breaking down numbers and using known multiplication facts to speed up the process.
In mental math, using the distributive property is a common way to simplify. You distribute one number over the parts of the other number which have been broken down, thus transforming one difficult problem into several smaller, solvable ones. Simplifying mental math problems requires practice but once mastered, it significantly speeds up your ability to handle mathematical calculations. Remember to always start with the largest components first to reduce the overall number of steps, streamline the process, and reach your solution faster.