Problem 61 - Algebra 1: Concepts and Skills

Question

Rewrite the expression with positive exponents. $$ \left(6 x^{-3}\right)^{3} $$

Step-by-Step Solution

Answer

The expression \((6x^{-3})^3\) rewritten with positive exponents is \(216/x^{27}\)

Step 1: Recall the Power of a Power Rule

The Power of a Power Rule states that \((a^m)^n = a^{m*n}\). Meaning, when you raise a power to a power, you multiply the exponents. In this expression, \(a = 6x^{-3}\) and \(n = 3\). However, because \(a = 6x^{-3}\) is a product, we will apply the rule to each part of the product separately.

Step 2: Apply the Power of a Power Rule

Applying the Power of a Power Rule, we have \((6)^{3}*(x^{-3})^{3}\). Calculating the power of \(6\) and \(-3\) separately, we have \(6^3 = 216\) and \((-3)^3 = -27\). Therefore, the expression becomes \(216x^{-27}\).

Step 3: Convert to Positive Exponent

The expression \(216x^{-27}\) is equivalent to \(216*(1/x^{27})\) as per the rule of negative exponents. According to the negative exponent rule, \(a^{-n} = 1/a^n\). Therefore, with this conversion, our final expression with all positive exponents will be \(216/x^{27}\).

Key Concepts

Negative Exponents

Negative exponents can seem tricky at first, but they follow a simple rule. When you see a negative exponent, it's a way of expressing the reciprocal of a number raised to the opposite positive exponent. For example, if you have a term like \(x^{-3}\), it's the same as saying \(1/x^3\).
To make this concept more digestible, think of a negative exponent as an instruction to "flip" the base element. So, \(a^{-n}\) becomes \(1/a^n\).

• It helps simplify expressions, especially when combined with other rules.
• Converting negative exponents to positive is crucial for clarity and easier computation.

This concept is relied upon when dealing with complex equations and algebraic expressions, ensuring calculations remain consistent and manageable.

Power of a Power Rule

The Power of a Power Rule is invaluable when simplifying expressions where a power is raised to another power. It states that \((a^m)^n = a^{m*n}\). This means, for any base, you multiply the exponents to simplify the expression.
Let's take a closer look at how it works with the example \((6x^{-3})^3\).

• First, separate the components: \(6\) and \(x^{-3}\).
• Apply the rule to each: \(6^3 = 216\) and \((x^{-3})^3 = x^{-9}\).

You'll notice this method maintains the base's integrity while manipulating the exponent values, allowing us to manage even the most complicated algebraic expressions effectively.

Exponentiation

Exponentiation is the process of raising a number to a power, which is fundamental in higher mathematics. It involves the repeated multiplication of a number by itself, defined by the exponent.
In our example, \(6^3\) represents \(6\times6\times6 = 216\). Likewise, \((x^{-3})^3\) uses the negative exponent in replication, but requires special handling via the power of a power rule.
Key points about exponentiation:

• The base number is multiplied by itself as many times as the exponent indicates.
• It's a compact way of expressing large multiplicative series.
• Exponentiation obeys specific rules like the power of a power rule, which helps in simplifying expressions.

Exponentiation is a foundational skill that aids in exploring more advanced mathematical territories, including calculus and algebraic transformations.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and arithmetic operations like addition, subtraction, multiplication, and division. They're used to represent mathematical concepts universally.
In an expression like \((6x^{-3})^3\), you're dealing with a product of numbers and variables raised to powers or exponents.

• They often contain terms, where each term is a product of a constant and variables raised to exponents.
• Understanding how to manipulate terms, using rules such as handling negative exponents and applying power rules, is crucial.

Algebraic expressions can look complex, but breaking them into parts and analyzing each part's role helps in simplifying and understanding them. This simplification paves the way for solving equations and modeling real-world scenarios.