Question

Write each expression as a power raised to a power. There may be more than one correct answer. $$ 2^{3 x} $$

Short answer

Question: Rewrite the expression \(2^{3x}\) as a power raised to a power. Answer: \(2^{(3^1 \cdot x^1)}\)

Step-by-step solution

Identify the given expression

The given expression is \(2^{3x}\).

Use the exponent rule

Since we want to express \(3x\) as a power raised to a power, we can consider that \(3x=3(x)\). In this case, \(3x\) can be re-written as $$3^1 \cdot x^1$$ because \(3x\) is the same as \(3 \cdot x\) and \(3=3^1\), \(x = x^1\).

Rewrite the expression

Now that we know that \(3x\) can be expressed as $$3^1 \cdot x^1$$, we can rewrite the given expression as $$2^{(3^1 \cdot x^1)}$$. So the given expression can be written as a power raised to a power as, $$ 2^{(3^1 \cdot x^1)} $$

Key concepts

power of a power

The concept of a "power of a power" in mathematics involves expressing one power raised to another power. When we encounter an expression like \((a^m)^n\), it indicates that the base \(a\) is being raised to the power of \(m\), and then that entire expression is raised to the power of \(n\). The notation simplifies to \(a^{m \cdot n}\).
This rule helps us simplify complex expressions by reducing them to a simpler form with fewer exponential components.
In the given exercise, \(2^{3x}\) can be understood as first expressing \(3x\) as a product of powers \((3^1 \cdot x^1)\), and then the expression \(3x\) becomes the power to which the base 2 is raised. So, rather than multiplying the powers directly, the exercise rephrases \(3x\) for comprehension and illustrates how to use these expressions efficiently.

exponent rules

Exponent rules are a set of guidelines used to simplify expressions involving powers. They make calculations involving exponents much more manageable. Let's look at a few key rules:

• **Product of Powers Rule**: If you multiply expressions with the same base, you add the exponents: \(a^m \times a^n = a^{m+n}\).
• **Power of a Power Rule**: As explained earlier, when raising an exponent to another power, you multiply the powers: \((a^m)^n = a^{m \cdot n}\).
• **Power of a Product Rule**: When raising a product to a power, you raise each factor to the power: \((ab)^n = a^n b^n\).

In the step-by-step solution, the use of \(3^1 \cdot x^1\) illustrates these rules effectively. It shows how to break down complex algebraic terms into simpler powers, which can then be used to express the entire expression as a power.

algebraic expressions

Algebraic expressions are combinations of variables, numbers, and operations. They form the foundation of algebra and help us model real-world problems with mathematical language. An expression can include terms like \(3x\) or \(2a^2b\).
In these expressions, combining numbers with variables through multiplication or other operations is essential. The original exercise, \(2^{3x}\), is an algebraic expression where exponential rules apply to both numbers and variables.
Understanding how these expressions work is crucial. When expressions are written as a power, it allows us to simplify and manipulate them to solve equations or model scenarios efficiently.
Rewriting \(3x\) as \(3^1 \cdot x^1\) showcases the flexibility of algebraic expressions and the power of exponentiation in mathematics.