Problem 24 - Algebra 2

Question

Simplify. $$ -\sqrt[6]{x^{6}} $$

Step-by-Step Solution

Answer

The simplified form is \(-x\).

Step 1: Understand the Problem

We need to simplify the expression \(-\sqrt[6]{x^{6}}\). This involves finding the simplified form of a sixth root.

Step 2: Simplify the Sixth Root

The expression \(\sqrt[6]{x^6}\) can be simplified by recognizing that taking the sixth root of \(x^6\) is effectively reversing the operation of raising \(x\) to the sixth power. This results in \(x\).

Step 3: Apply the Negative Sign

Notice that our original expression is \(-\sqrt[6]{x^{6}}\). After simplifying the sixth root to \(x\) in Step 2, we apply the negative sign, resulting in \(-x\).

Step 4: Final Simplified Expression

Putting it all together, the simplified form of \(-\sqrt[6]{x^{6}}\) is \(-x\).

Key Concepts

Understanding the Sixth Root

When we talk about roots, including the sixth root, we are looking to reverse the effect of raising a number or variable to a power. The concept of a "sixth root" for an expression like \(x^6\) means finding a number, that when raised to the 6th power, gives us exactly the original number, here \(x^6\). In mathematical terms, for a sixth root of \(x^6\), you can visualize it like this: \(\sqrt[6]{x^6} = x\).
This happens because the operations of raising to a power and taking a root are inverses of each other.

• The sixth root notation \(\sqrt[6]{}\) means the inverse operation of raising to the 6th power.
• Thus, \(\sqrt[6]{x^6} = x\) because we are taking the root of a perfect power.

When simplifying, always be aware of whether the result is positive or negative based on the context of the problem and the variable being used.

Applying the Negative Sign

In mathematics, the negative sign is used to denote the opposite of a number. In our original expression, \( -\sqrt[6]{x^{6}} \), the negative sign indicates the opposite of whatever result we obtain from \( \sqrt[6]{x^{6}} \). This is crucial in ensuring accuracy during simplification.

• First, simplify the expression \( \sqrt[6]{x^{6}} = x \).
• Next, apply the negative sign, which results in \( -x \).

A common misunderstanding here could be the placement of the negative sign—it's not the sixth root of \( -x^6 \), but rather the negative outside the root operation. Order of operation matters here!

Using Exponent Laws

Exponent laws are a set of rules that simplify expressions involving powers of numbers and variables. These laws are vital for understanding how we can manage roots and powers effectively and accurately. In our context with \(x^6\), we can make a connection between powers and roots.

• One key principle is that taking a root of a power like \(x^6\) is simply the inverse operation, bringing you back to the base \(x\).
• According to exponent laws, \((x^m)^n = x^{m\cdot n}\); when \(n\) is a fraction, it signifies a root instead of a further power.

Thus, \(\sqrt[6]{x^6}\) simplifies as \(x^{6\cdot \frac{1}{6}} = x^1 = x\). Exponent laws help ensure that you perform operations correctly while recognizing how powers and roots are related.