Question

Write each expression as a power raised to a power. There may be more than one correct answer. $$ 4^{2 \cdot 4} $$

Short answer

Question: Rewrite the expression \(4^{2 \cdot 4}\) as a power raised to a power. Answer: \((2^2)^8\)

Step-by-step solution

Analyze the given expression

We have the expression \(4^{2 \cdot 4}\). Our goal is to rewrite this in the form \((a^b)^c\).

Rewrite the base as a power raised to a power

We know that \(4\) can be written as \(2^2\). So, we can substitute this in our expression: $$ (2^2)^{2 \cdot 4} $$

Simplify the exponent

Now we have \((2^2)^{2 \cdot 4}\). We need to simplify the exponent \(2 \cdot 4\): $$ (2^2)^8 $$

Write the final answer

After rewriting and simplifying the expression, we get the final answer: $$ (2^2)^8 $$ This is one possible way to write the given expression as a power raised to a power.

Key concepts

Powers of Numbers

In mathematics, powers of numbers are a way to represent repeated multiplication of a number by itself. When you see a number like \(4^3\), it means the base number \(4\) is multiplied by itself three times: \(4 \times 4 \times 4\). This is where understanding the base and the exponent becomes crucial.

• **Base:** The number that is being multiplied.
• **Exponent:** Indicates how many times the base is multiplied by itself.

Understanding powers is essential for simplifying complex mathematical expressions. Especially in algebra, using powers expedites calculations by replacing repeated multiplication with a single power notation. This is efficient and reduces the chance of computational errors.

Simplifying Expressions

Simplifying expressions is a key skill in algebra. It involves reducing a complex expression into its simplest form while keeping the expression equivalent in value. This often makes the expression easier to work with and analyze. When simplifying expressions with powers, the **power raised to a power rule** is particularly handy. It states: when a power is raised to another power, you multiply the exponents. For instance, in the expression \((2^2)^8\), instead of working out each step by writing it down repeatedly, you multiply the two exponents (2 and 8) to find \(2^{16}\).

• Start with the given expression.
• Identify parts of the expression that can be simplified using power rules.
• Apply the rules to combine or reduce the terms.

Simplifying not only makes equations more manageable but also unveils deeper properties that can be more readily understood or used for further calculations.

Mathematical Notation

Mathematical notation is a universally recognized language for writing mathematical expressions. It allows mathematicians and students to communicate ideas precisely and concisely. Understanding and using mathematical notation is key to mastering mathematics. In the context of powers, the notation \(a^b\) stands for "a raised to the power of b," where \(a\) is the base and \(b\) is the exponent. This notation provides clarity and uniformity in expressions involving exponential terms. By rewriting an expression like \(4^{2 \cdot 4}\) as \((2^2)^8\), we use notation to convey the same value in a more structured manner.

• Learn common symbols like ^ for exponents and \(\times\) for multiplication.
• Practice reading and writing complex expressions using proper notation.
• Check for parentheses which indicate the order of operations.

Proper use of notation helps prevent misunderstandings and errors, and it ensures that calculations are interpreted uniformly across different contexts.