Problem 25 - Algebra 1

Question

Write your answer as a power or as a product of powers. $$ \left(7^{4}\right)^{2} $$

Step-by-Step Solution

Answer

The answer in form of a power of power to \( \left(7^{4}\right)^{2} \) is \(7^{8}\).

Step 1: Identifying the rule of exponents applicable

In the exercise, we have a number \(7^{4}\) raised to another power, \(2\), the formula \( (a^m)^n = a^{m*n} \) applies. In our case, \(a=7\), \(m=4\), \(n=2\) which means we have to multiply the inner exponent \(4\) by the outer exponent \(2\).

Step 2: Applying the rule

When we apply the rule, we get: \(7^{4*2}\) which means we need to multiply the exponents \(4\) and \(2\) together.

Step 3: Evaluate the new exponent

Multiplication gives us \(7^{8}\), which signifies that 7 is being multiplied by itself eight times.

Key Concepts

Rules of Exponents

Exponents play a significant role in algebra as they represent how many times a number, called the base, is multiplied by itself. Understanding the rules of exponents is key to simplifying expressions and solving equations involving exponential terms. Here are a few fundamental rules that you need to know:

• Product Rule: When multiplying powers with the same base, keep the base and add the exponents, represented as \( a^m \cdot a^n = a^{m+n} \).
• Quotient Rule: When dividing powers with the same base, keep the base and subtract the exponents, \( \frac{a^m}{a^n} = a^{m-n} \), given \( n \) is not zero.
• Power of a Power: When raising an exponent to another exponent, multiply the exponents, shown as \( (a^m)^n = a^{m \cdot n} \).
• Zero Exponent Rule: Any nonzero base raised to the power of zero equals one, \( a^0 = 1 \).
• Negative Exponent Rule: A negative exponent represents the reciprocal of the base raised to the positive exponent, \( a^{-n} = \frac{1}{a^n} \).

When solving problems with exponents, carefully identify which rule applies to each part of the expression before performing the operation.

Exponential Notation

In algebra, exponential notation is used to write repeated multiplication of the same number in a concise form. It consists of two parts: the base and the exponent. The base is the number being multiplied, and the exponent, which appears as a small number above and to the right of the base, tells you how many times to multiply the base by itself.
For example, \( 5^3 \) means that you multiply 5 by itself three times: \( 5 \times 5 \times 5 \), which equals 125. Exponential notation becomes particularly handy when dealing with very large numbers. Instead of writing out \( 10 \times 10 \times 10 \times 10 \times 10 \), we can simply write \( 10^5 \), greatly simplifying expressions and calculations. Remember that the base can be any real number, and the exponent can be positive, negative, or zero, each resulting in different calculations based on the rules of exponents.

Power of a Power Property

The power of a power property in algebra simplifies expressions where an exponent is raised to another exponent. This rule states that when we have an expression such as \( (a^m)^n \), we should multiply the exponents together to get \( a^{m \cdot n} \).
Let's consider an example based on the exercise: \( (7^4)^2 \). According to the power of a power rule, we multiply the inner exponent (4) by the outer exponent (2), resulting in \( 7^{4 \cdot 2} \) or \( 7^8 \). This means that the number 7 is used as a factor eight times when it's expanded. It is crucial to note that this rule only applies to exponents with the same base. This property makes it easier to handle large numbers and solve equations more efficiently by reducing the steps needed to simplify exponential expressions.