Finding the Cube Root of a Product of Negative Numbers

The question asks for the cube root of the product of -64 and -1331. This can be written as $\sqrt[3]{-64 \times -1331}$.

First, let's calculate the product of the two numbers:

$-64 \times -1331$

When two negative numbers are multiplied, the result is a positive number.

$-64 \times -1331 = 64 \times 1331$

Now we need to find the cube root of this positive product, which is $\sqrt[3]{64 \times 1331}$.

We can use the property of cube roots that states the cube root of a product is the product of the cube roots: $\sqrt[3]{a \times b} = \sqrt[3]{a} \times \sqrt[3]{b}$.

Applying this property, we get:

$\sqrt[3]{64 \times 1331} = \sqrt[3]{64} \times \sqrt[3]{1331}$

Next, we find the cube root of each number:

• To find $\sqrt[3]{64}$, we look for a number that, when multiplied by itself three times, equals 64. $4 \times 4 \times 4 = 16 \times 4 = 64$. So, $\sqrt[3]{64} = 4$.
• To find $\sqrt[3]{1331}$, we look for a number that, when multiplied by itself three times, equals 1331. $10 \times 10 \times 10 = 1000$, and $11 \times 11 \times 11 = 121 \times 11 = 1331$. So, $\sqrt[3]{1331} = 11$.

Now, we multiply the individual cube roots:

$\sqrt[3]{64} \times \sqrt[3]{1331} = 4 \times 11$

$4 \times 11 = 44$

Alternatively, we could first find the cube root of each negative number and then multiply.

• $\sqrt[3]{-64} = -4$, because $(-4) \times (-4) \times (-4) = 16 \times (-4) = -64$.
• $\sqrt[3]{-1331} = -11$, because $(-11) \times (-11) \times (-11) = 121 \times (-11) = -1331$.

The cube root of the product is then:

$\sqrt[3]{-64 \times -1331} = \sqrt[3]{-64} \times \sqrt[3]{-1331} = (-4) \times (-11)$

$(-4) \times (-11) = 44$

Both methods give the same result.

Therefore, the cube root of -64 × -1331 is 44.