Problem 66 - Algebra 2

Question

Simplify. $$ \sqrt[4]{(-8)^{4}} $$

Step-by-Step Solution

Answer

8

Step 1: Understand the Expression

The expression is \( \sqrt[4]{(-8)^4} \). We are asked to find the fourth root of \((-8)^4\). This means we need to determine what number, when raised to the fourth power, gives us \((-8)^4\).

Step 2: Evaluate \((-8)^4\)

To evaluate \((-8)^4\), calculate \((-8) \times (-8) \times (-8) \times (-8)\). Begin by calculating \((-8) \times (-8) = 64\). Then, calculate \(64 \times (-8) = -512\), and finally, \(-512 \times (-8) = 4096\). Thus, \((-8)^4 = 4096\).

Step 3: Find the Fourth Root

Now that we have \((-8)^4 = 4096\), the expression becomes \( \sqrt[4]{4096} \). We need to determine which number raised to the fourth power results in 4096. Since \(8^4 = 4096\), the fourth root of 4096 is 8.

Step 4: Conclusion

Thus, the fourth root of \((-8)^4\), which simplifies the initial expression \(\sqrt[4]{(-8)^4}\), is 8.

Key Concepts

Fourth Roots

A root is the inverse operation of exponentiation and finding the fourth root of a number involves determining which number, when raised to the power of four, equals the given value. The notation \( \sqrt[4]{x} \) is used to denote the fourth root of a number \( x \). In simple terms, if you have \( \sqrt[4]{4096} \), you are trying to find a number that results in 4096 when multiplied by itself four times.

The key point is understanding that raising a number to a power is the opposite of finding its root. For example, \( 8^4 = 4096 \), so \( \sqrt[4]{4096} = 8 \).

• Fourth root notation: \( \sqrt[4]{x} \)
• Inverse of exponentiation
• Used to break down numbers raised to the fourth power

By embracing these concepts, you simplify expressions like \( \sqrt[4]{(-8)^4} \) effectively, narrowing it down to a simpler number.

Exponentiation

Exponentiation is a mathematical operation involving two numbers, the base and the exponent. The exponent indicates how many times the base multiplies by itself. In the expression \( (-8)^4 \), -8 is the base and 4 is the exponent. This means -8 is multiplied by itself four times, i.e., \( (-8) \times (-8) \times (-8) \times (-8) \).

Let's break down exponentiation:

• The base determines the initial value
• The exponent shows how many times to repeat the base as a factor
• For example, \( a^n \) means multiply \( a \) by itself \( n \) times

Understanding exponentiation helps you evaluate and simplify expressions like \( (-8)^4 \). It creates large numbers but also offers a structured way to tackle complex operations.

Properties of Exponents

Understanding the properties of exponents makes working with expressions easier and reveals patterns. Some helpful rules include the power of a power property, the product of powers, and the quotient of powers.

### Key Properties:

• Power of a power: \( (a^m)^n = a^{m \cdot n} \)
• Product of powers: \( a^m \cdot a^n = a^{m+n} \)
• Quotient of powers: \( \frac{a^m}{a^n} = a^{m-n} \)

By relying on these properties, you can transform complex looking problems into simpler forms. For example, by understanding that \( (-8)^4 \) is simply a repeated multiplication of \(-8\) and utilizing the power property, you can evaluate such expressions confidently. Properties like these lay the foundation for solving algebraic problems efficiently.