Problem 7 - 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 simplest form is \(-9c^{8}\).

Step 1: Understand the Operation

The expression given is \(-\left(3c^{4}\right)^{2}\). This involves a negative sign and requires us to square the term \(3c^{4}\).

Step 2: Apply the Power Rule

Using the power rule, \((ab)^{n} = a^{n}b^{n}\), we can expand \(\left(3c^{4}\right)^{2}\) as \(3^{2}\cdot\left(c^{4}\right)^{2}\).

Step 3: Simplify Each Part

Calculate each part: \(3^{2} = 9\) and \((c^{4})^{2} = c^{8}\) using the power of a power rule \((x^{m})^{n} = x^{mn}\).

Step 4: Combine Results and Apply Negative Sign

Combine the simplified parts to get \(9c^{8}\). Apply the negative sign from the original expression: the result is \(-9c^{8}\).

Step 5: Verify Result is Simplified

Check if \(-9c^{8}\) is in its simplest form. Since it cannot be simplified further, this is the final answer.

Key Concepts

Exponent Rules

Exponent rules are essential tools for simplifying expressions involving powers. They help us handle the operations on numbers and variables raised to exponents. Here are the key exponent rules you should know:

• **Product of powers rule:** When multiplying like bases, add their exponents. For example, \(x^m \cdot x^n = x^{m+n}\).
• **Quotient of powers rule:** When dividing like bases, subtract the exponents: \(\frac{x^m}{x^n} = x^{m-n}\), provided \(x eq 0\).
• **Zero exponent rule:** Any base with an exponent of zero equals one, \(x^0 = 1\), as long as \(x eq 0\).

These rules provide a methodical way to handle expressions and reduce complexity by operating on the exponents directly.
Understanding these rules lays a foundation for more complex operations such as the power of a power rule.

Power of a Power

The power of a power rule is a useful tool when simplifying algebraic expressions that involve exponents. When you have an expression like \((x^m)^n\), the power of a power rule tells you to multiply the exponents: \((x^m)^n = x^{m \cdot n}\). This rule simplifies the process of dealing with nested exponential terms.For instance, in the expression \((c^4)^2\), you apply the power of a power rule by multiplying the exponents \(4\) and \(2\) to get \(c^{4 \cdot 2} = c^8\).
When you encounter a compound term such as \((3c^4)^2\), apply the rule separately to both the coefficient and the variable, like this:

• Calculate the coefficient squared: \(3^2 = 9\).
• Multiply the exponents of the variable: \((c^4)^2 = c^8\).

Then, you can combine these results in the simplified form: \(9c^8\).
Mastering this rule enables you to tackle more complicated algebraic expressions with ease.

Negative Sign in Expressions

Negative signs in expressions can significantly change the outcome and need careful handling. In mathematics, the presence of a negative sign before an expression implies that all terms within it should be multiplied by -1.In the exercise, we dealt with the expression \(-\left(3c^4\right)^2\).
When the negative sign is outside the parentheses, it means that the entire result obtained from the operation should be negated. Here's how it works:

• First, resolve the expression within the parentheses: \(\left(3c^4\right)^2 = 9c^8\).
• Then apply the negative sign to the result: \(-1 \cdot 9c^8 = -9c^8\).

Remember, ignoring the negative sign can lead to incorrect results. Always keep track of it, especially in longer and more complex calculations.
By being mindful of the negative sign, you ensure that the expression remains accurate as you simplify it.

Simplifying Algebraic Expressions

Simplifying algebraic expressions means reducing them to their simplest form. This often involves combining like terms, factoring, and using the rules of exponents. The aim is to make the expression as straightforward as possible, without changing its value.Let's break down the process:

• **Combine like terms:** Terms with the same variable and exponent can be added or subtracted together.
• **Use exponent rules:** To simplify powers, apply rules like product, quotient, or power of powers as needed.
• **Factor where possible:** Look for common factors that can be taken out of the expression to simplify it further.

In the given exercise with the expression \(-9c^8\), we find that it's already in its simplest form.
There are no like terms to combine, no further factors to extract, and the operations within it are fully resolved. Recognizing when an expression is fully simplified is as crucial as being able to simplify it.