Which expression is equivalent to 16^3

Equivalent Expression

We start with the expression $16^3$ and aim to find an equivalent power of 2.

First, note that $16$ can be written as a power of 2:

Thus,

Using the power of a power rule $(a^m)^n = a^{m \cdot n}$:

Therefore, the expression equivalent to $16^3$ is:

Explanation

The power of a power rule is a property of exponents that simplifies expressions where one exponent is raised to another. This rule is particularly useful in algebra and calculus for simplifying complex exponential expressions.

The Rule

In mathematical terms, the power of a power rule states that:

This means that when you have a base $a$ raised to an exponent $m$, and this entire expression is then raised to another exponent $n$, you can multiply the exponents $m$ and $n$ to get the simplified form.

Example

Consider the expression $(3^{2})^{4}$:

So, $(3^{2})^{4}$ simplifies to $3^{8}$.

Why It Works

To understand why this rule works, let's expand the expression $(a^{m})^{n}$ manually:

Application

This rule is particularly useful in:

• Simplifying algebraic expressions
• Solving exponential equations
• Calculating higher-order derivatives in calculus where exponents are involved

By applying the power of a power rule, you can significantly reduce the complexity of calculations involving exponents, making it easier to manipulate and solve equations.

Understanding the Concept

The conversion of a base to a power of 2 involves transforming numbers from one base to another where the target base is a power of 2, such as binary (base 2), quaternary (base 4), or hexadecimal (base 16). This can often simplify certain computational tasks and understand numerical representations in digital systems.

Step-by-Step Conversion

1. Divide the Number into Groups: When converting from a smaller base to a larger power of 2, start by dividing the number into groups. The size of each group is determined by the power of 2. For instance, if converting from binary to hexadecimal, group the binary digits into sets of 4 (since $16 = 2^4$).

2. Convert Each Group: Next, convert each group from the original base to the target base. For example, convert each group of 4 binary digits to a single hexadecimal digit.

Example: Binary to Hexadecimal

Let's convert the binary number $110110111001$ to hexadecimal:

1. Grouping: Break the binary number into groups of 4:

   $$1101\ 1011\ 1001$$

2. Conversion:

   $$1101_2 = D_{16},\quad 1011_2 = B_{16},\quad 1001_2 = 9_{16}$$

So, $110110111001_2 = DB9_{16}$.

Benefits of Conversion

• Simplification: Converting to bases that are powers of 2 can make logical operations simpler and more intuitive.
• Error Checking: Many error-detection and correction algorithms are easier to implement in powers of 2 bases.
• Storage and Memory: Digital systems often operate more efficiently in binary or hexadecimal, aiding in memory addressing and data representation.

Mathematical Representation

For a more formal transformation between bases, consider a number $N$ in base $b_1$ being converted to base $b_2$ where $b_2 = 2^k$:

To convert $N$ to base $b_2$:

1. Express $N$ in Base 10:

   $$N_{10} = d_n \cdot b_1^n + d_{n-1} \cdot b_1^{n-1} + \ldots + d_0 \cdot b_1^0$$

2. Convert $N$ from Base 10 to Base $b_2$:

   $$N_{b_2} = \sum_{i=0}^{m} e_i \cdot b_2^i$$

where $e_i$ are the digits in the base $b_2$ representation.

By following these steps, you can efficiently convert numbers between bases, especially those where the target is a power of 2, which is beneficial for digital and computational tasks.