Make Sense? In Exercises 119–122, determine whether each statement makes sense or does not make sense, and explain your reasoning. ____ ⁴√(−8)⁴ cannot be positive 8 because the power and the index cancel each other.

Verified step by step guidance

1. Step 1: Understand the expression $\sqrt[4]{(-8{)}^4}$. This is the fourth root of $(-8{)}^4$.
2. Step 2: Calculate $(-8{)}^4$. When you raise a negative number to an even power, the result is positive. So, $(-8{)}^4=4096$.
3. Step 3: Now, find the fourth root of 4096, which is $\sqrt[4]{4096}$.
4. Step 4: The fourth root of a positive number is also positive, so $\sqrt[4]{4096}=8$.
5. Step 5: Conclude that the statement does not make sense because $\sqrt[4]{(-8{)}^4}$ simplifies to 8, not negative 8.

Key Concepts

Radical and Exponent Rules

In algebra, the rules governing radicals and exponents are crucial for simplifying expressions. The expression ⁴√(−8)⁴ involves both a radical (the fourth root) and an exponent (to the fourth power). According to these rules, raising a negative number to an even power results in a positive number, while taking the fourth root of a positive number yields a non-negative result.

Even and Odd Powers

Understanding the distinction between even and odd powers is essential in algebra. An even power of a negative number results in a positive value, while an odd power retains the negative sign. In the case of (−8)⁴, since 4 is even, the result is positive 4096, which is relevant when evaluating the fourth root of this expression.

Properties of Roots

The properties of roots dictate how to handle expressions involving radicals. Specifically, the nth root of a number is defined as the value that, when raised to the nth power, yields the original number. For example, ⁴√(4096) equals 8, as 8⁴ equals 4096. This property is critical in determining the validity of the statement regarding the fourth root of (−8)⁴.