Question

$$ \text { If } 3^{a}=w, \text { express } 3^{3 a} \text { in terms of } w \text { . } $$

Short answer

Question: Express \(3^{3a}\) in terms of \(w\), given that \(3^a = w\). Answer: \(w^3\)

Step-by-step solution

Write the Power Rule

Recall the power rule for exponents which states that \((a^m)^n = a^{mn}\). We will apply this rule to rewrite \(3^{3a}\).

Rewrite \(3^{3a}\) using the power rule

Applying the power rule to \(3^{3a}\), we get: $$(3^a)^3=3^{3a}$$

Replace \(3^a\) with \(w\)

Given that \(3^a = w\), we can now substitute this into our expression from Step 2: $$(w)^3=3^{3a}$$

Simplify the expression

The expression is now simplified to: $$w^3 = 3^{3a}$$ So, \(3^{3a}\) can be expressed as \(w^3\).

Key concepts

Power Rule

The power rule is a crucial concept when dealing with exponents. It helps to simplify expressions that involve repeated multiplication. According to the power rule, if you have an exponent raised to another exponent, you multiply them together. Specifically, \[(a^m)^n = a^{mn}\] This means when a base with an exponent is raised again by another exponent, you simply multiply both exponents to find the solution.
An example will show this more clearly: if you have \[(3^2)^4\] using the power rule gives you \[3^8\]. This concept is very useful in simplifying expressions because instead of calculating powers separately, you can use this rule to directly calculate the resulting power by multiplication.

Exponents

Exponents are a way of expressing repeated multiplication of the same number by itself. For example, \[3^2\] means you multiply 3 by itself, resulting in \[3 imes 3 = 9\].
The number being multiplied is called the base, and the exponent tells you how many times to multiply that base by itself.
Here are some key points about exponents:

• A positive exponent means you multiply the base by itself the number of times indicated.
• A zero exponent always results in 1, regardless of the base (assuming the base is not zero).
• A negative exponent indicates division or a reciprocal. For example, \[3^{-2} = \frac{1}{3^2} = \frac{1}{9}\].

Understanding these rules is essential in algebra, especially when simplifying and solving equations, as they form the foundation for more complex mathematical concepts.

Algebraic Expressions

In algebra, expressions involve a combination of numbers, variables, and operations (such as addition, subtraction, multiplication, and division). They do not have an equals sign, making them different from equations.
Variables are symbols, like \(x\) or \(a\), which represent unknown values or quantities that can vary.
An example of an algebraic expression is:\[3a + 2\]This means the variable \(a\) is multiplied by 3 and then 2 is added.

• To simplify algebraic expressions, you need to apply operations while respecting the rules of arithmetic and algebra.
• Look for like terms or common factors which can be combined or factored out to make the expression simpler.
• Expressions can be a single term or multiple terms linked by operators.

Recognizing and manipulating these forms is fundamental to solving mathematical problems and developing logical reasoning skills.