Problem 62 - College Algebra Essentials

Question

Evaluate each expression in Exercises \(55-66,\) or indicate that the root is not a real number. $$\sqrt[4]{(-2)^{4}}$$

Step-by-Step Solution

Answer

The fourth root of (-2)^4 is 2

Step 1: Evaluate (-2)^4

Start by evaluating the expression inside the bracket. When -2 is raised to the power of 4, the negative sign becomes positive because the power is even. Therefore, (-2)^4 equals 16.

Step 2: Evaluate the fourth root of 16

The fourth root of 16 can be calculated as the number that multiplied by itself four times equals 16. The number 2 satisfies this condition, since 2*2*2*2 equals 16.

Key Concepts

Fourth Roots

The concept of a fourth root is similar to that of a square root, but instead of looking for a number that, when multiplied by itself twice, gives the original number, we look for a number that, when multiplied by itself four times, gives the original number. This can be represented mathematically as \( \sqrt[4]{x} \), where \( x \) is the original number.

• To find the fourth root of a number, think of "undoing" the process of raising a number to the power of four.
• For example, if we have the expression \( \sqrt[4]{16} \), we are looking for a number that multiplied by itself four times equals 16.
• Through trial or by recognizing powers, we see that 2 is such a number, since \( 2 \times 2 \times 2 \times 2 = 16 \).

Learning how to find roots, including fourth roots, is a useful skill in algebra that helps students understand the inverse operations of exponentiation. This process aids in both simplifying expressions and solving equations.

Exponentiation

Exponentiation is a mathematical operation involving two numbers: the base and the exponent. The base is the number being multiplied, and the exponent, also known as the power, tells us how many times the base is multiplied by itself.

• In the expression \((-2)^4\), \(-2\) is the base, and 4 is the exponent.
• The exponent 4 indicates that \(-2\) should be multiplied by itself four times: \((-2) \times (-2) \times (-2) \times (-2)\).
• Notice that multiplying an even number of negative numbers together results in a positive product, which is why \((-2)^4 = 16\).

Understanding exponentiation helps in solving problems involving powers and roots, and it's crucial for simplifying algebraic expressions and solving exponential equations. It's a fundamental building block for advanced mathematical concepts.

Imaginary Numbers

Imaginary numbers come into play when dealing with the square roots of negative numbers. A standard real number squared will never result in a negative product, which is where imaginary numbers become valuable.

• If you try to find the square root of a negative number, like \( \sqrt{-4} \), you will need to use imaginary numbers, since there is no real number that can achieve this.
• Imaginary numbers are based on \( i \), which is defined as the square root of \(-1\).
• Therefore, \( \sqrt{-4} \) is expressed as \( 2i \), since \( (2i)^2 = 4 \times -1 \).

Imaginary numbers expand the realm of number systems, allowing mathematicians and scientists to solve equations and undertake calculations that involve the square roots of negative numbers. While not involved directly in the original step-by-step solution, knowing about imaginary numbers is important for understanding the limitations and extensions of real numbers.