Problem 68 - Algebra 1: Concepts and Skills

Question

Evaluate the expression. \(\left(3^{3}\right)^{2}\)

Step-by-Step Solution

Answer

The value of the expression \( \left(3^{3}\right)^{2} \) is 729.

Step 1: Identify the Base and Exponents

In this expression \( \left(3^{3}\right)^{2} \), 3 is the base (a) and 3 and 2 are the exponents (m and n respectively).

Step 2: Apply the Power of a Power Rule

Apply the power of a power rule, which states: \( (a^{m})^{n} = a^{mn} \). Here, a is 3, m is 3 and n is 2. So it becomes \(3^{(3\times2)} = 3^{6}\).

Step 3: Compute the Power

Compute \(3^{6}\) to find the final answer. The number 3 multiplied by itself 6 times equals 729.

Key Concepts

Power of a Power Rule

The power of a power rule is a key concept in exponentiation that simplifies handling expressions with multiple exponents. When you have an expression in the format \((a^m)^n\), this rule helps you combine the exponents into one. According to this rule:

• Multiply the exponents \(m\) and \(n\).

This means the expression \((a^m)^n\) becomes \(a^{mn}\).
In our example problem \((3^3)^2\), the base is 3 and the exponents are 3 and 2. Using the power of a power rule, we multiply these exponents: \(3 \times 2 = 6\). Therefore, \((3^3)^2 = 3^6\). This simplifies what can look complex at first into something much easier to manage.

Base and Exponent Concepts

Understanding base and exponent concepts is essential when working with powers. The base in an exponential expression is the number being multiplied.
The exponent shows how many times the base multiplies itself.

• In \(3^3\), 3 is the base.
• The exponent 3 indicates that the base should be multiplied by itself three times: \(3 \times 3 \times 3\).

In multi-layered exponents like \((3^3)^2\), the outer exponent refers to how many times we must process the inner power of base 3. This knowledge is fundamental for evaluating and simplifying any exponential expression.

Evaluating Expressions

Evaluating expressions involves calculating the actual number from the given exponential terms. Once the base and exponents are properly understood, and rules like the power of a power are applied, solving the expression becomes straightforward. In the expression \((3^3)^2\), using the power of a power rule gives us \(3^6\). To evaluate this, compute \(3 \times 3 \times 3 \times 3 \times 3 \times 3\).
It might be easier to first calculate smaller parts, like:

• \(3^2 = 9\), then \(9^2 = 81\), followed by \(81 \times 3 = 243\), and again \(243 \times 3 = 729\).

So, \(3^6\) evaluates to 729, which is the final solution for the expression. Breaking it down like this can simplify and clarify the process. This way, whether numbers are small or large, you have a method to methodically approach the computation.