Problem 9 - Algebra and Trigonometry

Question

\(7-28\) Evaluate each expression. $$ \left(2^{3}\right)^{2} $$

Step-by-Step Solution

Answer

64

Step 1: Understand the Expression

The expression given is \((2^3)^2\). This is a power of a power problem, where we have a base \(2\), an initial exponent \(3\), and an outer exponent \(2\). In exponentiation, when raising a power to another power, we multiply the exponents.

Step 2: Apply the Power of a Power Rule

According to the power of a power rule, \(a^{m^n} = a^{m \cdot n}\). Therefore, we multiply the exponents in the given expression: \((2^3)^2 = 2^{3 \times 2}\). This simplifies to \(2^6\).

Step 3: Calculate the Power

Now, we need to calculate \(2^6\). This involves multiplying the base, \(2\), by itself 6 times: \(2 \times 2 \times 2 \times 2 \times 2 \times 2 = 64\). Thus, \(2^6 = 64\).

Step 4: Summarize the Solution

We have simplified and calculated the expression. The final answer to \((2^3)^2\) is \(64\).

Key Concepts

Power of a Power Rule

Exponentiation can seem tricky when dealing with multiple layers of exponents. However, understanding the 'power of a power rule' makes it much easier to handle. This rule states that when you have a number with an exponent raised to another exponent, you multiply the exponents together. For example, in the expression \((a^m)^n\), instead of performing the calculation anew, you can simplify it using the rule to get \(a^{m \cdot n}\).
This turns what might seem like a complex problem into a straightforward multiplication task. Remember, the 'power of a power' rule is very helpful in breaking down complex exponent expressions into simpler parts.

Multiplying Exponents

When applying the power of a power rule, the focus is on multiplying exponents. In the example \((2^3)^2\), we identify that we need to multiply the exponent 3 by the exponent 2. However, it's crucial to remember this only applies to situations where a power is raised to another power (like \((a^m)^n\)).
To multiply these exponents, you simply perform the multiplication: \(3 \times 2 = 6\). Therefore, the expression becomes \(2^6\). This process cuts down the steps by eliminating the need to work out elaborate calculations directly. It’s a mathematical shortcut that saves time and reduces complexity.

Calculating Powers

Once we have simplified using the power of a power rule, we arrive at a simpler expression that needs calculation. Here, \(2^6\) becomes the expression we need to evaluate. Calculating powers s simple if you remember that it’s just repeated multiplication of the base. So, \(2^6\) is \(2 \times 2 \times 2 \times 2 \times 2 \times 2 = 64\).
Breaking it down step-by-step:

• First two 2s: \(2 \times 2 = 4\)
• Next pair: \(4 \times 2 = 8\)
• Follow by: \(8 \times 2 = 16\)
• Then: \(16 \times 2 = 32\)
• Finally: \(32 \times 2 = 64\)

This demonstrates how the multiplication builds up to get our final result. Each time, we're simply doubling the number, leading to the exponential growth as shown.