Problem 3 - Algebra 2

Question

Multiply, if possible. Then simplify. $$ \sqrt[3]{9} \cdot \sqrt[3]{-81} $$

Step-by-Step Solution

Answer

-9

Step 1: Multiply the cubic roots

The multiplication is as follows \( \sqrt[3]{9} \cdot \sqrt[3]{-81} = \sqrt[3]{9 \times -81} \).

Step 2: Compute the product

Compute the product inside the cubic root: \( \sqrt[3]{9 \times -81} = \sqrt[3]{-729} \)

Step 3: Simplify the Cubic Root

The cubic root of -729 is -9, because \( (-9)^3 = -729 \).

Key Concepts

Simplifying Radicals

Simplifying radicals is a way of reducing expressions to their simplest form while keeping their value intact. In the context of cubic roots, simplifying involves finding smaller numbers whose cubes match the given value.
Consider the example of the cubic root \(\sqrt[3]{-729}\). To simplify:

• Find a number which, when multiplied by itself three times, equals \(-729\): This number is \(-9\) because \( (-9) \times (-9) \times (-9) = -729\).
• This simplification results in a cleaner, more manageable number.

Simplifying radicals helps in understanding and working with complex expressions. It allows us to express results using simpler numbers, which makes further calculations much easier.

Multiplying Radicals

When multiplying radicals, especially cubic roots, it's important to understand that we're essentially combining the inside values before simplifying.
For example, \(\sqrt[3]{9} \cdot \sqrt[3]{-81}\) means you need to multiply the numbers within: \(9 \times -81\).
This results in a single new cubic root: \(\sqrt[3]{-729}\).

• Always multiply the numbers under the radicals first.
• Once multiplied, simplify the new radical if possible.

Breaking down the process into steps helps retain clarity. This ensures that you don't accidentally alter the original value by improper multiplication or skipping simplification steps.

Integer Exponents

Integer exponents express how many times to use a number in a multiplication.
In our context, the exponent of 3 in a cubic root tells us that a number is used thrice in multiplication to achieve the given result. \( (-9)^3 = -729 \) illustrates this concept:

• Here, \(-9\) is multiplied by itself three times.
• This results in \(-729\), linking the radical simplification process with integer exponents.

Understanding integer exponents is key to mastering radicals, as they show the deep interconnection between multiplication and root extraction.