Problem 60 - Elementary and Intermediate Algebra

Question

Divide, if possible, and check. If a quotient is undefined, state this. $$ \frac{-64}{-7} $$

Step-by-Step Solution

Answer

The quotient is approximately 9.14.

Step 1: Simplify the signs

Since both the numerator and the denominator are negative, dividing a negative number by another negative number gives a positive result. This means the expression \(\frac{-64}{-7}\) can be rewritten as \(\frac{64}{7}\).

Step 2: Perform the division

Divide 64 by 7 to get the quotient. \(\frac{64}{7} \approx 9.14\). So, the result of \(\frac{64}{7}\) is approximately 9.14.

Step 3: Check the division

Multiply the divisor (7) by the quotient (approximately 9.14) to check the division. \( 7 \times 9.14 \approx 64 \). Since multiplication yields a number close to the original numerator, the division was performed correctly.

Key Concepts

negative numbers

Negative numbers are integers that are less than zero. They are located to the left of zero on the number line. When you divide a negative number by another negative number, the result is always positive. This is because the two negative signs cancel each other out. For example, in the division \(\frac{-64}{-7}\), both numbers are negative. Therefore, the result is positive, giving us \(\frac{64}{7}\). This understanding is crucial when dealing with operations involving negative numbers, making sure to always consider the signs of the numbers involved.

division

Division is the process of determining how many times one number is contained within another. In this exercise, we need to divide 64 by 7. First, simplify the signs if necessary. Then perform the division. Here, \(\frac{64}{7}\) gives 9.14 approximately. To check the division, multiply the divisor (7) with the quotient (9.14). \(\text{7} \times 9.14 \approx 64\), indicating that the division was done correctly. Remember to always check your results to ensure accuracy.

simplifying fractions

Simplifying fractions is the process of reducing a fraction to its simplest form. This involves ensuring that both the numerator and denominator are as small as possible and that they do not share any common factors other than 1. For example, in our problem, \(\frac{64}{7}\) is already in its simplest form because 7 is a prime number and does not divide 64. Simplifying fractions makes them easier to work with and understand, especially in more complex mathematical operations.