Problem 34 - College Algebra

Question

For the following exercises, simplify each expression. $$ 3 \sqrt[3]{-432}+\sqrt[1]{16} $$

Step-by-Step Solution

Answer

The simplified expression is -2.

Step 1: Simplify the Cube Root

The first step involves solving \( \sqrt[3]{-432} \). The cube root of a negative number is negative. \( \sqrt[3]{-432} \) is \( -\sqrt[3]{432} \). Breaking 432 down gives us \( 432 = 2^4 \times 3^3 \). So, \( \sqrt[3]{432} = \sqrt[3]{2^4 \times 3^3} = 2 \times 3 = 6 \). Thus, \( \sqrt[3]{-432} = -6 \).

Step 2: Solve the Cube Root Expression

Now substitute the cube root value into the original expression: \( 3 \times -6 = -18 \). This simplifies the cube root component to \( -18 \).

Step 3: Simplify the Square Root

Now address the square root \( \sqrt[1]{16} \). However, \( \sqrt[1]{16} \) is just \( 16 \) because any number to the power of one is itself.

Step 4: Combine the Results

Finally, combine the simplified results from both parts: \( -18 + 16 \). Simplifying gives us: \( -18 + 16 = -2 \).

Key Concepts

Cube Root

A cube root is a value that, when multiplied by itself three times, produces the original number. For example, the cube root of 27 is 3 because when you multiply 3 × 3 × 3, you get 27.
Now, if you have a negative number under the cube root, such as \(-432\), the result will be negative. This is because a negative number times itself twice more still results in a negative product.
For example:

• The cube root of \(-8\) is \(-2\), because \(-2 imes -2 imes -2 = -8\).
• In our exercise, \(-432\) can be factored into \(2^4 imes 3^3\). So the cube root of \(-432\) simplifies as \(-6\).

Therefore, understanding the factorization of numbers helps simplify cube roots.

Square Root

The square root gives you a number that, when multiplied by itself, equals the original number. For example, the square root of 16 is 4 because 4 × 4 = 16.
In our problem, the expression \(\sqrt[1]{16}\) is simplified as just 16. The notation \(\sqrt[1]{x}\) simply represents the number itself. It's because any number to the first power remains unchanged.
Here’s a quick breakdown:

• \(\sqrt[1]{25}\) equals 25 because it’s 25 raised to the power of 1.
• In the exercise: \(\sqrt[1]{16}\) = 16.

Understanding this can prevent any confusion regarding the notation of square roots involving exponents.

Negative Numbers

Negative numbers provide a method for representing quantities less than zero. They can be handled in operations like addition, subtraction, and multiplication.
When you multiply a positive number by a negative number, your result is negative. For instance:

• 3 × \(-4\) equals \(-12\).
• In the problem, \(3 \sqrt[3]{-432}\) leads to 3 × \(-6\) which results in \(-18\).

Also, combining a negative number with a positive number requires attention, as seen in our exercise:
\(-18 + 16\) results in \(-2\).
Thus, careful attention to sign will allow correct simplification of expressions.

Exponent Properties

Understanding exponent properties is crucial for simplifying complex expressions. Different properties explain how numbers behave under exponents.

• **Product of Powers**: \((a^m imes a^n = a^{m+n})\)
• **Power of a Power**: \((a^{m})^n = a^{m imes n}\)
• **Power of a Product**: \((ab)^m = a^m imes b^m\)

In the context of cube and square roots, knowing exponent properties aids in simplification. For instance, when solving \(432 = 2^4 imes 3^3\), it's beneficial to see how it breaks down neatly under the cube root. It allows us to quickly realize that \(\sqrt[3]{2^4 imes 3^3}= 2 imes 3 \), making the simplification process easier.
Understanding these rules can simplify and solve expressions accurately.