Problem 80 - Algebra A Combined Function

Question

Find each power of \(i\). $$ (5 i)^{4} $$

Step-by-Step Solution

Answer

The result of \((5i)^4\) is \(625\).

Step 1: Recall Powers of i

Remember that the imaginary unit \(i\) has specific powers with repeated cycles: \(i^1 = i\), \(i^2 = -1\), \(i^3 = -i\), and \(i^4 = 1\). Any power of \(i\) can be reduced by recognising this cycle.

Step 2: Simplify the Expression

The expression \((5i)^4\) can be expanded using the property of exponents. This results in \((5^4) \times (i^4)\). This allows us to consider the components separately.

Step 3: Calculate \(5^4\)

Compute \(5^4\) as follows: \(5^4 = 5 \times 5 \times 5 \times 5 = 25 \times 5 = 125 \times 5 = 625\).

Step 4: Evaluate \(i^4\)

From Step 1, we know that \(i^4 = 1\).

Step 5: Combine Results

Combine the components from Steps 3 and 4. Multiply \(625\) (from \(5^4\)) by \(1\) (from \(i^4\)): \[ (5i)^4 = 625 \times 1 = 625 \]

Step 6: Provide the Final Answer

The final result of \((5i)^4\) is simply \(625\).

Key Concepts

Complex Numbers

Complex numbers are an essential concept in mathematics, particularly in fields involving algebra and calculus. A complex number is defined as a number that can be expressed in the form of \(a + bi\), where \(a\) and \(b\) are real numbers, and \(i\) is the imaginary unit. The part \(a\) is the real component, and \(b\) multiplied by \(i\) is the imaginary component. Complex numbers allow for the expansion and generalization of real number operations and their applications.

• Real part: \( a \)
• Imaginary part: \( b \)

This flexibility makes complex numbers powerful tools in solving equations that have no solution within the realm of real numbers alone. For example, consider the equation \(x^2 + 1 = 0\), which has no real solution, but it does have complex solutions, \(x = i\) and \(x = -i\). By combining these real and imaginary components, mathematicians and scientists can better model and understand complex phenomena.

Imaginary Unit

The imaginary unit \(i\) is fundamental when dealing with complex numbers. It is defined by the property \(i^2 = -1\). This definition helps solve equations that otherwise couldn't be addressed with real numbers alone. Imaginary numbers and the imaginary unit provide solutions that elegantly extend our numeric system.Here are the basic powers of \(i\) to keep in mind:

• \(i^1 = i\)
• \(i^2 = -1\)
• \(i^3 = -i\)
• \(i^4 = 1\)

The powers of \(i\) repeat every four steps, creating a cycle that is easy to remember. This cyclical nature is useful when calculating higher powers of \(i\). For instance, as shown in the solved exercise, knowing that \(i^4 = 1\) simplifies the expression \((5i)^4\) by reducing \(i^4\) to \(1\), turning the complex expression into a simple scalar multiplication.

Exponents and Powers

Understanding exponents and powers is crucial when working with both real and complex numbers. Exponents provide a shorthand way of expressing repeated multiplication. For example, \(5^4\) means multiplying 5 by itself four times, resulting in 625. In complex numbers, this principle extends to include the imaginary unit \(i\).When dealing with powers of complex numbers such as \((5i)^4\), you break it down to simpler parts:

• Calculate \(5^4\) to get 625.
• Resolve \(i^4\) using the known cycle to get 1.

This simplification harnesses the repetitive cycles of the imaginary unit. Using these cycles, powers of \(i\) become easy to handle, allowing you to focus on solving more complex equations. By mastering exponents and understanding their patterns, especially those involving the imaginary unit, you cultivate a stronger grasp of both elementary and advanced mathematical concepts.