Problem 124 - College Algebra Essentials

Question

What difference is there in simplifying \(\sqrt[3]{(-5)^{3}}\) and \(\sqrt[4]{(-5)^{4} ?}\)

Step-by-Step Solution

Answer

The simplification \(\sqrt[3]{(-5)^{3}}\) results in -5, while the simplification \(\sqrt[4]{(-5)^{4}}\) yields +5.

Step 1: Simplify Cube Root Expression

Start with the cube root expression \(\sqrt[3]{(-5)^{3}}\). With cube roots or any odd roots, the negative sign stays. The cube root of a value cubed is the original value, but the negative sign stays. Therefore, the simplification results in -5.

Step 2: Simplify Fourth Root Expression

Now, examine the fourth root expression \(\sqrt[4]{(-5)^{4}}\). With fourth roots or any even roots, the result is always positive. This is because when you square (or raise to the fourth, etc.) any negative number, the result is always positive, and the root returns the positive square root (or fourth root, etc.) . Hence, the simplification results in +5.

Key Concepts

Understanding Cube Roots

When it comes to simplifying radical expressions involving cube roots, it's essential to grasp what cube roots represent. A cube root, denoted by \( \sqrt[3]{x} \), asks the question: 'What number multiplied by itself three times equals x?' For instance, in the expression \( \sqrt[3]{(-5)^3} \), you're essentially looking for the number that, when cubed, gives you \( -5 \times -5 \times -5 \).

One critical aspect of cube roots is that they can return negative answers. Because we are dealing with an odd number of multiplications of the same negative number, the result retains its negativity. Thus, simplifying \( \sqrt[3]{(-5)^3} \) straightforwardly gives you -5, maintaining the negative sign of the initial expression.

• Cube roots can have negative results.
• The cube root of a number cubed returns the original number.
• Negative signs remain with odd roots.

Fourth Roots and Even Radicals

Conversely, with fourth roots, symbolized as \( \sqrt[4]{x} \), the scenario changes significantly. The fourth root of a number is asking, 'Which number raised to the fourth power equals x?' An interesting property of even radicals, like the fourth root, is that they always yield a non-negative result. This stems from the fact that multiplying an even number of negative terms together always results in a positive product.

For the expression \( \sqrt[4]{(-5)^4} \), each -5 is multiplied together four times. The negatives cancel each other out, leaving us with a positive result, thus when we take the fourth root, we're left with a positive value. Therefore, simplifying \( \sqrt[4]{(-5)^4} \) leads to +5 – the absolute value of the original base number -5.

• Fourth roots of a number to the fourth power will be positive.
• Even roots result in non-negative numbers.
• This property applies for all even-numbered roots.

Negative Number Roots

When simplifying roots of negative numbers, the parity of the root—whether it is odd or even—plays a pivotal role. For odd roots, such as cube roots, the rule is straightforward: the root of a negative number is also negative. This is because an odd number of negative factors will produce a negative product.

However, the situation is quite different with even roots. Mathematically, you cannot have a real even root of a negative number because you can't multiply a number by itself an even number of times to get a negative result. Therefore, when dealing with expressions like \( \sqrt{(-x)} \) where x is a positive number, if the root is an even number, the result would be considered non-real or imaginary in the realm of complex numbers.

• Odd roots of negative numbers remain negative.
• Even roots of negative numbers result in imaginary numbers (not covered in real number systems).
• The parity of the root (odd or even) determines the outcome.