Problem 32 - Algebra 1: Concepts and Skills

Question

Write the expression as a single power of the base. \(\left(7^{4}\right)^{2}\)

Step-by-Step Solution

Answer

The expression \((7^{4})^{2}\) written as a single power of the base is \(7^{8}\).

Step 1: Identify the Base and Powers

Here, the base is 7, the first power is 4, while the second power is 2, so we can write the expression as \((7^{4})^{2}\). Now the problem can be approached by using the power of a power rule.

Step 2: Apply the Power of Power Rule

Now apply the power of a power rule, which is \((a^m)^n = a^{m*n}\), replace \(a\) with 7, \(m\) with 4 and \(n\) with 2. This gives \(7^{4*2}\). Multiply the powers.

Step 3: Simplify the Expression

Multiplying the powers 4 and 2 together, we get \(7^{8}\). This is our final result.

Key Concepts

Power of a Power Rule

When dealing with exponents, the **power of a power rule** is an essential tool to simplify expressions. This rule states that when you have an exponent raised to another exponent, you multiply the exponents together. For example,

• If you have an expression like \((a^m)^n\), you apply the power of a power rule to write it as \(a^{m \cdot n}\).

In the context of our exercise, we see this with \((7^4)^2\). By applying the rule, multiply 4 and 2 together to get \(7^{4 \cdot 2} = 7^8\). This transformation simplifies the expression, making it easier to understand and solve.
Understanding the power of a power rule helps in breaking down complex exponential expressions into simpler forms, which can be crucial in both basic algebra and advanced math problems. Remember, the key is in the multiplication of exponents when dealing with a power raised to another power.

Simplifying Expressions

Simplifying expressions involves making them as uncomplicated as possible, while retaining the essential structure and values. In the example given, \((7^4)^2\), it might appear complex, but using rules such as the power of a power rule makes it simpler. By applying the rule, we reduce the expression to \(7^8\).
The objective of simplifying is to reach a format that is easy to interpret and use in other calculations. This process often involves moving from a "nested" or "tiered" form of expression to a more direct form. In mathematical terms, simplifying can mean performing basic operations to reduce the number of terms, like multiplying the powers in our case.

• You'll find that simplified forms are easier to compare with other similar expressions.
• They often use less space and computational resources when being processed.

Thus, mastering expression simplification is a fundamental skill in math, enabling clearer communication and comprehension of mathematical ideas.

Base and Exponent Concept

The **base and exponent concept** is a cornerstone of understanding exponential expressions. In any expression like \(a^b\),

• \(a\) is the base, the number that is being multiplied.
• \(b\) is the exponent, indicating how many times the base is used as a factor.

In the problem given, the base is 7, which is the number repeatedly multiplied by itself. The first exponent given is 4, which tells us to multiply 7 by itself four times. The expression \((7^4)^2\) then uses another exponent, 2, to show that this process must happen again, effectively doubling the amount of times we multiply 7 by itself due to the power of a power rule.
Recognizing the base and exponent is vital as it sets the foundation for all work with exponents. It allows you to apply rules like the power of a power or to simplify expressions correctly. Understanding this concept at a deeper level enriches your ability to interpret more complex algebraic equations and hence is fundamental for anyone studying math at any level.