Problem 69 - Algebra 1: Concepts and Skills

Question

In Exercises \(65-70,\) simplify. Then use a calculator to evaluate the expression. $$ \left(2^{2} \cdot 2^{2}\right)^{2} $$

Step-by-Step Solution

Answer

So the simplified and evaluated version of the expression \( (2^{2} \cdot 2^{2})^{2} \) is 256.

Step 1: Expand the expression

We start off by expanding the expression, replacing the power of a power with multiplication. The expression becomes \( 2^2 \cdot 2^2 \cdot 2^2 \cdot 2^2 \).

Step 2: Apply the rules of indices

Here, we apply the rule of indices \( a^{m} \cdot a^{n} = a^{m + n} \) because the base of the power is the same (2). Adding all the exponents together gives us \( 2^8 \).

Step 3: Calculate the power

Now, we calculate the power \( 2^8 \) using a calculator or doing manual multiplication, which equals to 256.

Key Concepts

Powers of a Power

Understanding the concept of 'powers of a power' is essential when simplifying exponential expressions. It involves taking an exponent to another exponent. The basic rule is quite straightforward: when you raise a power to another power, you multiply the exponents. For example, \( (a^m)^n = a^{m \times n} \). In the offered exercise, we have \( (2^2)^2 \). When we apply the 'powers of a power' rule, we multiply the exponents: 2 and 2, resulting in \( 2^4 \), which further simplifies the expression.

Rules of Indices

The 'rules of indices' also known as the 'laws of exponents', are a set of rules for operating with exponential terms. Key concepts include \( a^m \cdot a^n = a^{m+n} \), when multiplying powers with the same base, and \( \frac{a^m}{a^n} = a^{m-n} \), when dividing powers with the same base. In our exercise, we use the multiplication rule to combine \( 2^2 \cdot 2^2 \cdot 2^2 \cdot 2^2 \) into \( 2^8 \), by adding the exponents, illustrating how indices can simplify multiplication of exponential terms.

Evaluating Expressions

Evaluating expressions is the process of simplifying or finding the value of algebraic or numerical expressions. When evaluating exponential expressions, we must adhere to the correct order of operations and apply the appropriate rules of indices. After we've simplified \( 2^8 \) by combining the expressions as mentioned in the rules of indices, we evaluate it by calculating the value, which could be achieved using basic manual multiplication—multiplying the base number, two, by itself eight times—or more efficiently using a calculator, yielding the value 256.

Exponential Notation

Exponential notation is a concise way to represent repeated multiplication of the same number. It is written as a base number raised to a power or exponent (for example, \( a^n \) where \( a \) is the base and \( n \) is the exponent). This notation is particularly useful for representing very large or very small numbers efficiently and is essential in many areas of mathematics and science. The given exercise \( (2^2)^2 \) is a classic example of exponential notation being used to denote a more complex multiplication process in a simple form.