Problem 59 - Algebra 2

Question

Tell whether each equation is true for all, some, or no values of the variable. Explain your answers. $$ \sqrt{x^{6}}=x^{3} $$

Step-by-Step Solution

Answer

The equation \(\sqrt{x^{6}}=x^{3}\) holds true for any positive number and zero, but fails for any negative numbers. Thus, the equation is true for some values of \(x\), not all.

Step 1: Analyze the Properties of Exponents and Radicals

First, you need to recognize that \(x^{6}\) is effectively \(x^{3} * x^{3}\) and that the square root of any number, including \(x^{6}\), is a number that multiplies by itself to give \(x^{6}\). In this case, that number would be \(x^{3}\). Therefore, we may reasonably assume that the equation \(\sqrt{x^{6}}=x^{3}\) is accurate.

Step 2: Consider Positive and Negative Variables

However, you also need to take into consideration that because the square root symbol (radical) by default denotes only the positive root, \(\sqrt{x^{6}}\) will always give a positive outcome or zero. This is different from \(x^{3}\), which will yield negative values when \(x\) is a negative number (as a negative number raised to an odd power is negative).

Step 3: Finalize the Conclusion

Therefore, when \(x\) is a positive number or zero, the equation \(\sqrt{x^{6}}=x^{3}\) holds true (this covers infinite values of \(x\)). But, when \(x\) is a negative number, the equation does not hold true. Thus, the equation is correct for some, but not all, values of \(x\).

Key Concepts

Understanding Radicals

Radicals are mathematical symbols that denote roots of numbers. The most common radical is the square root, represented by the symbol \( \sqrt{} \). When you see \( \sqrt{x^{6}} \), it means you need to find a number that you can multiply by itself to get back to \( x^{6} \). In simpler terms, if \( x^{6} = y \times y \), then \( \sqrt{x^{6}} = y \). In our example, \( y \) turns out to be \( x^{3} \), since \( x^{3} \times x^{3} = x^{6} \).

• Radicals simplify numbers to their root values.
• The square root only returns the principal (positive) root.

This means while \( \sqrt{x^{6}} \) seems equivalent to \( x^{3} \) when \( x \) is positive or zero, it ignores negative values because it does not account for the negative root.

Positive and Negative Numbers

Understanding the behavior of positive and negative numbers is crucial, especially when dealing with exponents and radicals. Positives and negatives have different rules:

• When you multiply two positive numbers, the result is positive.
• When you multiply two negative numbers, the result is also positive.
• However, a negative number raised to an odd power will remain negative.

This is the key difference in how \( \sqrt{x^{6}} \) and \( x^{3} \) behave. When \( x \) is negative, \( x^{3} \) will also be negative because negative numbers raised to odd powers stay negative. Conversely, the square root \( \sqrt{x^{6}} \) will not turn negative, as radicals primarily provide positive results. This discrepancy is why the equation \( \sqrt{x^{6}}=x^{3} \) only holds true for positive numbers and zero, not negatives.

Equation Validity

Equation validity explains when and why an equation holds true. To determine this, observe the properties involved, such as exponents and radicals, and their effects on positive and negative numbers. In this exercise, we're examining \( \sqrt{x^{6}}=x^{3} \):

• For \( x > 0 \): Both sides equal \( x^{3} \), confirming the equation.
• For \( x = 0 \): Both sides are zero, so the equation holds true.
• For \( x < 0 \): \( \sqrt{x^{6}} \) is positive, but \( x^{3} \) is negative, so the equation does not hold.

Thus, the equation is only valid for some values of \( x \), specifically non-negative values. By examining each case, we confirm the equation applies only under specific conditions. This analysis highlights the importance of verifying equations through consideration of all potential variable values.