Question

\(\left(6 a^{3}\right)^{3}-(2 b)^{4}+c^{-2}=?\) (A) \(6 a^{9}-2 b^{4}+\frac{1}{c^{2}}\) (B) \(18 a^{9}-8 b^{4}+\frac{1}{c^{2}}\) (C) \(21 a^{6}-16 b^{4}+\frac{1}{c^{2}}\) (D) \(216 a^{9}-16 b^{4}+\frac{1}{c^{2}}\)

Short answer

Option (D): \(216 a^{9} - 16 b^{4} + \frac{1}{c^{2}}\) is correct.

Step-by-step solution

Simplifying the First Term

We need to simplify the expression \((6a^{3})^{3}\). Use the power of a power property: \((x^m)^n = x^{m \cdot n}\). Apply this here: \((6a^{3})^{3} = 6^3 \cdot (a^3)^3 = 6^3 \cdot a^{3 \cdot 3} = 216a^9\).

Simplifying the Second Term

Now simplify \((2b)^{4}\). Use the same power of a power property: \((2b)^{4} = 2^4 \cdot b^4 = 16b^4\).

Simplifying the Third Term

For the term \(c^{-2}\), recall the property: \(x^{-n} = \frac{1}{x^n}\). Thus, \(c^{-2} = \frac{1}{c^2}\).

Combine All Simplified Terms

Now combine the results from all steps: \(216a^9 - 16b^4 + \frac{1}{c^2}\).

Identify the Correct Option

Finally, compare the expression we derived, \(216a^9 - 16b^4 + \frac{1}{c^2}\), with the options. The expression matches option (D).

Key concepts

Understanding Exponents

Exponents are a shorthand method to indicate that a number needs to be multiplied by itself a certain number of times. For example, \(a^3\) implies that \(a\) is multiplied by itself three times: \(a \cdot a \cdot a\). When dealing with exponents, it's essential to know several key properties that help simplify expressions.

• Product of Powers: If you multiply two exponents with the same base, you simply add the exponents. So \(a^m \cdot a^n = a^{m+n}\).
• Power of a Power: When raising an exponent to another power, you multiply the exponents. Thus, \((x^m)^n = x^{m \cdot n}\).
• Power of a Product: When raising a product to a power, apply the power to each factor inside. For example, \((xy)^n = x^n \cdot y^n\).
• Negative Exponents: A negative exponent indicates reciprocation. So \(x^{-n} = \frac{1}{x^n}\).

Mastering these rules will greatly assist in simplifying complex expressions and is crucial for tackling algebraic problems.

Grasping Algebraic Expressions

Algebraic expressions are mathematical phrases that can involve numbers, variables, and operations. Variables are symbols, often letters, used to represent numbers. In an expression like \(6a^3\), the terms include a coefficient (6), a variable base \(a\), and an exponent (3).

• Coefficients are the numerical part of the terms. They multiply the variables and can be positive, negative, integers, or fractions.
• Variables represent unknown or changeable numbers and are at the heart of algebra. They are crucial when forming equations to solve problems.
• Operators such as addition (+) and subtraction (-) are used to combine different terms into more complex expressions.

Each term in an expression can be simplified separately and then combined to give a cleaner, more compact solution. Understanding how to manipulate algebraic expressions is essential for solving equations and performing various algebraic operations.

Mastering Simplification

Simplification is the process of reducing an expression to its simplest form, making it easier to work with. When simplifying, the goal is to transform the problem into a more manageable form while maintaining its original value.

• Combine Like Terms: Look for terms with the same variable and exponent, and combine them by adding or subtracting their coefficients.
• Simplify Powers: Apply the rules of exponents to reduce terms, as shown in expressions like \((6a^3)^3\), which simplifies to \(216a^9\).
• Reduce Fractions: If the expression involves changes like \(c^{-2}\), transform it to \(\frac{1}{c^2}\) to simplify handling and clarity.
• Follow Arithmetic Operations: Ensure each step follows the correct order of operations: parentheses, exponents, multiplication and division, addition, and subtraction (PEMDAS).

Effective simplification enhances problem-solving speed and efficiency, making it an indispensable tool in math.