Problem 12 - Algebra 1

Question

Use the power of a power property to simplify the expression. $$ \left(2^{4}\right)^{3} $$

Step-by-Step Solution

Answer

The simplified expression of \( \left(2^{4}\right)^{3} \) is 4096

Step 1: Identify Base and Powers

The given expression is \( \left(2^{4}\right)^{3} \). Here, the base is 2 and the powers are 4 and 3.

Step 2: Apply the Power of a Power Rule

Using the property of exponents, the power of a power rule, \((a^{m})^{n} = a^{m*n}\), apply this rule to the given expression. Then the expression \(\left(2^{4}\right)^{3} \) reduces to \(2^{4*3} = 2^{12}\).

Step 3: Calculate the Final Value

Since \(2^{12}\) simply means that we multiply 2 by itself 12 times. So, \(2^{12} = 4096\)

Key Concepts

Power of a Power Property

When dealing with exponential expressions, the power of a power property is a useful tool to simplify them. This property specifically applies when you have an exponent raised to another exponent, like in \( (2^4)^3 \). Here, 4 and 3 are the exponents. The property states that \[ (a^m)^n = a^{m \times n} \] This means you multiply the exponents. For the expression \( (2^4)^3 \), using the power of a power property, you multiply 4 and 3 to get \( 2^{12} \). This property helps simplify complex expressions, making them easier to compute or use in further calculations.

Simplifying Expressions

Simplifying expressions means reducing them to their simplest form without changing their value. With exponents, this often involves using properties of exponents such as the power of a power property. In our example \( (2^4)^3 \), applying this property simplifies it to \( 2^{12} \). This transformation:

• Makes the expression easier to work with.
• Reduces potential for errors in computation.
• Provides a clearer understanding of the expression's actual value.

The final simplified form allows us to easily compute the value if needed, as \( 2^{12} = 4096 \). Simplifying is an important skill in algebra that aids both understanding and practical computation.

Exponential Notation

Exponential notation is a way of expressing numbers as a base raised to a power. This notation is compact and efficient, especially for dealing with large numbers. In \((2^4)^3\), both 4 and 3 are exponents, and 2 is the base. When simplified, it becomes \(2^{12}\), showing how the expression relates to powers of 2.Exponential notation conveys how many times a base is multiplied by itself:

• The base is the number being multiplied.
• The exponent indicates how many times the base appears in the multiplication.

For this example, \(2^{12}\) indicates multiplying 2 by itself 12 times, which gives a clear picture that the large number 4096 is a result of repeated multiplication. Exponential notation is not only for simplification but also emphasizes the operation process behind numbers.