Problem 62 - Elementary Algebra

Question

Write the following expressions using only positive exponents. Assume all variables are nonzero. $$ 3 y^{-3}(9 x) $$

Step-by-Step Solution

Answer

Question: Rewrite the expression 3y^{-3}(9x) using only positive exponents. Answer: The expression rewritten using only positive exponents is (27x)/(y^3).

Step 1: Rewrite the expression with a positive exponent for y

Applying the rule \((a^{-n}) = \frac{1}{a^{n}}\), we can rewrite \(y^{-3}\) as \(\frac{1}{y^3}\): $$ 3 \frac{1}{y^3} (9x) $$

Step 2: Simplify the expression

Now, combine the constants and the variables as follows: $$ \frac{3 \times 9x}{y^3} $$

Step 3: Multiply the constants and write the final expression

Multiply the constants 3 and 9 to get 27: $$ \frac{27x}{y^3} $$ The final expression with positive exponents is \(\frac{27x}{y^3}\).

Key Concepts

Algebraic Expressions

Algebraic expressions form the foundation of algebra, composed of numbers, variables, and operational symbols. In our exercise, we work with the expression \(3y^{-3}(9x)\). This expression includes:

• Coefficients: The numbers 3 and 9 accompanying the variables.
• Variables: Represented by symbols such as \(y\) and \(x\) that stand for unknown or changeable numbers.
• Exponents: In this case, the negative exponent \(y^{-3}\).

Understanding these components helps in manipulating and simplifying the expression. Every part plays a vital role, and together they represent a specific quantity or relation. Let's delve further into how we simplify and transform these parts while sticking to mathematical rules.

Exponent Rules

Exponent rules are a set of guidelines that dictate how to handle powers in algebraic expressions. A crucial rule we use is turning negative exponents into positive ones. The rule states that \(a^{-n} = \frac{1}{a^n}\). So, when faced with \(y^{-3}\), we convert it to \(\frac{1}{y^3}\).Other key exponent rules you might encounter include:

• Product of Powers: \(a^m \cdot a^n = a^{m+n}\)
• Power of a Power: \((a^m)^n = a^{m \cdot n}\)
• Power of a Product: \((ab)^n = a^n \cdot b^n\)

These rules simplify managing complex expressions. They ensure clarity and accuracy in deriving equivalent forms. While our primary focus was on negative exponents, knowing all exponent rules is beneficial for future algebraic tasks.

Simplifying Expressions

Simplifying expressions is about making them easier to understand or work with, without changing their value. It's about applying mathematical operations to condense expressions into simpler forms. In our exercise, we first rewritten negative exponents as positive. Then, we combine and simplify components.Steps in simplification include:

• Combining Constants: Multiply numbers together. Here, we multiplied 3 and 9 to get 27.
• Grouping Variables: Align them properly, using fresh or rewritten exponents.
• Using Fraction Form: Simplified negative exponents often involve placing variables in the denominator, leading to a fraction.

By following these steps, the original expression \(3y^{-3}(9x)\) transforms into the clear and concise \(\frac{27x}{y^3}\). This final form maintains equivalent value while adhering to rules and providing better readability.