Problem 78 - Algebra 2

Question

Simplify each radical expression. Use absolute value symbols as needed. $$ \sqrt[3]{-64 a^{81}} $$

Step-by-Step Solution

Answer

\(-4a^{27}\)

Step 1: Identify the coefficients and variables inside the cubic root.

The coefficient inside the cubic root is -64 and the variable is \(a^{81}\). We need to find the cubic roots of these two components.

Step 2: Determine the cubic root of -64.

The goal in this step is to find the cubic root of -64. Since the cube root of a negative number is also negative, we find the cube root of 64, which is 4, and keep the negative sign, getting -4.

Step 3: Determine the cubic root of \(a^{81}\).

We know that to get the cube root of a variable to a power, we divide the exponent by 3. Doing this for \(a^{81}\), we get \(a^{27}\).

Step 4: Combine the roots of the coefficient and the variable.

In the final step, we simply combine the results from steps 2 and 3 into a single expression. Therefore, the simplified version of \(\sqrt[3]{-64 a^{81}}\) is \(-4a^{27}\).

Key Concepts

Cubic Roots

Understanding cubic roots is essential when simplifying radical expressions. A cubic root asks you to find a number that, when multiplied by itself three times, gives the original number. For example, the cubic root of 8 is 2 because

• 2 × 2 × 2 = 8.

Cubic roots can also be taken from negative numbers. Unlike square roots, which result in imaginary numbers when dealing with negatives, cubic roots maintain the negative sign.
This is because a negative number multiplied by itself three times remains negative.
For instance, the cubic root of -64 is -4, since

• -4 × -4 × -4 = -64.

Cubic roots are important in algebra, enabling us to simplify expressions and solve equations involving higher powers.

Absolute Value

Absolute value represents the distance a number has from zero, always yielding a non-negative result. It is denoted by two vertical bars, like

• |x|.

This concept is particularly useful when simplifying expressions, as it helps to remove any negative signs when needed.
However, in the context of cubic roots and current exercise, it’s important to note that the absolute value isn’t necessary unless the problem specifically asks for only positive results.
For the expression does not require the use of absolute values because the context of the question involves roots which inherently manage negative results, like in the cube root case.

Variable Exponents

Simplifying expressions with variable exponents can seem tricky. But it's straightforward if you remember the exponent rules. When working with radicals like cubic roots, you divide the exponent by the root's degree.
For example, consider the term a^{81} in the expression.
To find its cube root, calculate

• 81 ÷ 3 = 27.

Therefore, a^{81}, when simplified, becomes a^{27}.
This process of breaking down the exponent simplifies complex variables and makes dealing with powers in algebra much easier.
Mastering these steps will help streamline solving equations with high powers.

Negative Numbers

Negative numbers often pose challenges in mathematical operations, but understanding their role is crucial when simplifying expressions. The beauty of math is that it provides rules that consistently lead to the correct results, whether numbers are positive or negative.
For instance, with cubic roots, if you start with a negative number, the result of the cube root will still be negative.
This happens because the multiplication of three negative numbers results in a negative.
So, for -64, the cubic root results in -4, since

• -4 x -4 x -4 = -64.

Understanding these core concepts helps ensure you correctly handle negative numbers in complex expressions.