Question

In Exercises \(5-8,\) rewrite the exponential expression to have the indicated base. \(16^{3 x}, \quad\) base 2

Short answer

Therefore, \(16^{3x}\) with a base of 2 is equal to \(2^{12x}\)

Step-by-step solution

Identify the new base for the exponential expression

Firstly, notice that the base of the expression should be rewritten from 16 to 2 while preserving the expression value. This can be done by expressing 16 as a power of 2, which is \(2^4\).

Substitute the base 16 with \(2^4\)

Now replace 16 in the expression with \(2^4\), which gives \((2^4)^{3x}\)

Apply the power of a power rule

The power of a power rule states that when raising a power to another power, the powers multiply together. So, \((2^4)^{3x}\) simplifies to \(2^{4\times3x}\) or \(2^{12x}\)

Key concepts

Power of a Power Rule

Understanding how to manipulate exponents is crucial in simplifying exponential expressions, and one essential principle to master is the power of a power rule. This rule comes into play when an exponential term is raised to another exponent—think of it as having a stack of powers. The rule states that when you have a base raised to an exponent, and this entire expression is then raised to another exponent, you multiply the two exponents together.

For example, if we take \( (a^m)^n \), we apply the power of a power rule by multiplying the exponents: \( a^{m \times n} \). This simplification step is a game-changer since it allows us to convert complex-looking expressions into a far more manageable form. It's like consolidating the strength of two repeated multiplications into a single, powerful punch!

Let's visualize this with a practical example from the exercise: \( (2^4)^{3x} \). Here, we multiply the inner exponent 4 with the outer exponent 3x, simplifying the expression to \( 2^{12x} \). This clarity not only makes the calculation smoother but also deepens our understanding of the overarching concepts of exponentiation.

Base Substitution in Exponents

Sometimes, you'll come across an exponential expression with a base that can be re-expressed as a smaller base raised to a power. This is where base substitution becomes a nifty tool. It involves replacing the original base with a new one while keeping the overall value of the expression unchanged. But why do this? Because it often simplifies the process of solving problems, especially when working with algebraic expressions involving exponents.

Take our example \(16^{3x}\), where we aim to rewrite it using base 2. Since 16 is a power of 2, specifically \(2^4\), we can substitute the base 16 with \(2^4\), thereby transforming the original expression into \( (2^4)^{3x} \). This step is skillfully using the relationships between numbers to pave the way for easier manipulation, which is especially helpful when handling more complex algebraic work.

Remember that choosing the right base for substitution depends on the given problem and what you're trying to achieve. It's like selecting the right tool for the job—it makes the work that much more efficient.

Simplifying Exponential Expressions

The goal of simplifying exponential expressions is to transform them into the most uncomplicated form possible, which can involve several steps, including applying the power of a power rule and base substitution, as we've seen. Simplification makes it easier to comprehend the expression and use it in further calculations.

Our primary strategies include identifying like bases, combining and reducing similar terms, and applying exponent rules. For instance, when we have terms like \(a^m \times a^n\), we can add the exponents since the bases are the same, resulting in \(a^{m+n}\). Conversely, when dividing exponential terms with the same base, we subtract the exponents (e.g., \(\frac{a^m}{a^n} = a^{m-n}\)).

In the case of the exercise, \(16^{3x}\) is already an exponential term, but we can simplify it by first expressing 16 as \(2^4\) and then applying the power of a power rule to end up with \(2^{12x}\). Simplification is like tidying up: it may involve several steps, but the result is a much cleaner and more usable space—metaphorically speaking, of course. Simplified expressions are essential for clearer understanding and easier handling in algebra.