Problem 62 - Elementary Algebra

Question

. \(\sqrt[3]{-27}\)

Step-by-Step Solution

Answer

-3

Step 1: Understand the Cube Root

To find the cube root of a number means to find a value that, when multiplied by itself three times, gives the original number.

Step 2: Identify the Number

In this case, you need to find the cube root of \(-27\). The negative sign indicates that the result will also be negative because the cube of a negative number is negative.

Step 3: Calculate

Determine what number multiplied by itself three times equals \(-27\). \(-3 \times -3 \times -3 = -27\).

Step 4: Result

Therefore, \(\root 3 \of -27 = -3\).

Key Concepts

cube root calculation

To calculate the cube root of a number, we look for a value that, when multiplied by itself three times, gives the original number. This is different from finding the square root, where we only multiply the number by itself twice.

For example, to find the cube root of 8, we look for a number that when multiplied three times equals 8. In this case, it is 2 because \[ 2 \times 2 \times 2 = 8 \].

The cube root symbol is \[ \sqrt[3]{} \], so the cube root of 8 is written as \[ \sqrt[3]{8} = 2 \]. When dealing with negative numbers, like in our exercise, the process is very similar but includes some additional considerations about negative values.

negative integers

A negative integer is any integer less than zero, such as -1, -2, -3, and so on. Negative integers follow specific rules when used in mathematical operations.

For example, they must be carefully managed in multiplication and other operations to ensure correct results. When dealing with cube roots, it’s important to understand that the cube root of a negative number is also negative.

This is because multiplying three negative numbers together always results in a negative product. As seen in our example, \[ -3 \times -3 \times -3 = -27 \]. Since the result is -27, the cube root of -27 is -3.

multiplication of negative numbers

Multiplying negative numbers can initially seem tricky, but it's straightforward once you know the rules. Here are some simple rules to follow:

• Negative \(\times\) Negative = Positive
• Negative \(\times\) Positive = Negative
• Positive \(\times\) Negative = Negative

For example, \[ -2 \times -3 \times -2 = -12 \]. Notice that in our exercise, we need the cube root of a negative number, resulting in three negative multiplications. This holds because \[ -3 \times -3 \times -3 = -27 \]. This consistently yields a negative result, so the cube root of a negative number will always be a negative number. This understanding is essential for solving problems involving cube roots and negative integers.