Problem 6 - Algebra 2 and Trigonometry

Question

Perform the indicated operations and write the result in simplest form. \(\left(-3 c^{4}\right)^{2}\)

Step-by-Step Solution

Answer

The simplified form is \(9c^8\).

Step 1: Apply the Power of a Power Property

When you raise a power to another power, you multiply the exponents. In this case, \((-3c^4)^2\) becomes \((-3)^2 imes (c^4)^2\).

Step 2: Calculate the Power of the Coefficient

Calculate \((-3)^2\), which means multiplying \(-3\) by itself. \(-3 imes -3 = 9\).

Step 3: Apply the Power of a Power to the Variable

For \(c^4\), when raised to the power of 2, you multiply the exponents: \(c^{4 imes 2} = c^8\).

Step 4: Write the Simplified Result

Combining the results from Steps 2 and 3, the expression \((-3c^4)^2\) simplifies to \(9c^8\).

Key Concepts

Understanding the Power of a Power Property

The Power of a Power Property is a key rule in exponentiation. It states that when you raise a power to another power, you multiply the exponents together. This property is vital for simplifying expressions involving exponents, allowing you to consolidate multiple powers into a single exponent. For example, in the expression \((c^4)^2\), the exponents 4 and 2 are multiplied together, resulting in \(c^{4 \times 2} = c^8\). This rule simplifies and streamlines computations, particularly in algebra, by reducing complex powers into manageable numbers. Remember, to apply this property, just keep the base the same and multiply the exponents.

The Process of Multiplying Exponents

Multiplying exponents is straightforward when using the Power of a Power Property and when dealing with expressions that have the same base. When you encounter an expression like \((c^m)^n\), the exponents are multiplied, resulting in \(c^{m \times n}\). This method helps compress lengthy expressions into simpler forms, making them easier to handle mathematically. When using negative bases, like \((-3)^2\), be careful to apply exponents correctly: this involves multiplying the number by itself, resulting in a positive value if the exponent is even, such as \((-3) \times (-3) = 9\). Multiplying exponents in this way helps you refine and simplify complicated algebraic expressions.

Simplifying Expressions to Their Simplest Form

Simplifying expressions involves reducing them to their simplest and most concise form without altering their value. This often requires the application of various algebraic properties, such as combining like terms and applying exponent rules like the Power of a Power Property. In the given exercise, we first applied the power property to \((-3c^4)^2\) to break it down into \((-3)^2\) and \((c^4)^2\). By calculating each part, \((-3)^2\) simplifies to 9 since it's the square of the base, and \((c^4)^2\) becomes \(c^8\). Finally, combine these to get the simplified form: \(9c^8\). By understanding how to apply these basic properties, simplifying becomes a straightforward task that enhances clarity and efficiency in problem-solving.