Problem 74 - Elementary Algebra

Question

Write \(\left(4 x^{3} y^{-4}\right)^{-2}\) so that only positive exponents appear.

Step-by-Step Solution

Answer

Question: Rewrite the expression \(\left(4 x^{3} y^{-4}\right)^{-2}\) with only positive exponents. Answer: \(\frac{y^8}{16x^6}\)

Step 1: Apply Power Rule to Exponents

Using the power rule \((a^m)^n = a^{mn}\), apply the exponent \(-2\) to both \(4\), \(x^3\), and \(y^{-4}\) inside the parentheses: \((4 x^{3} y^{-4})^{-2} = 4^{-2}(x^{3})^{-2}(y^{-4})^{-2}\).

Step 2: Simplify Exponents

Now, multiply the exponents, and simplify the expression: \(4^{-2}(x^{3})^{-2}(y^{-4})^{-2} = 4^{-2}x^{-6}y^8\).

Step 3: Convert Negative Exponents to Fractions

For any negative exponent, we apply the rule: \(a^{-m} = \frac{1}{a^m}\). Convert the negative exponents to their reciprocal forms: \(4^{-2}x^{-6}y^8 = \frac{1}{4^2}\cdot\frac{1}{x^6}\cdot y^8\).

Step 4: Simplify the Fraction

Now, simplify the fraction and multiply the terms together: \(\frac{1}{4^2}\cdot\frac{1}{x^6}\cdot y^8 = \frac{y^8}{16x^6}\). So, the expression \(\left(4 x^{3} y^{-4}\right)^{-2}\) with only positive exponents is: \(\frac{y^8}{16x^6}\).

Key Concepts

Power Rule

In the world of exponents, the power rule is a handy tool used to simplify expressions. The power rule states that when you have an exponent raised to another exponent, you can multiply the powers together. This can be expressed as \((a^m)^n = a^{mn}\).
For instance, if you start with \((x^3)^2\), using the power rule, you multiply the exponents: \(3 \times 2\), giving \(x^6\).
This rule helps in reducing complex exponentiation into a simpler form and is particularly useful in algebraic expressions involving powers.

Negative Exponents

Negative exponents can seem confusing at first, but understanding them is much simpler than it appears. When dealing with negative exponents, remember that they represent a reciprocal. The rule is: \(a^{-m} = \frac{1}{a^m}\). This means that any base with a negative exponent can be rewritten as one over that base with a positive exponent.

For example, \(x^{-3}\) becomes \(\frac{1}{x^3}\). This concept allows us to transform expressions so that all exponents become positive, making them easier to understand and solve.

Simplifying Expressions

Simplifying expressions is all about making them as straightforward as possible. This often involves using rules like the power rule and handling negative exponents. The goal is to rewrite the expression so that it's easier to work with while maintaining its value.

In our exercise, simplifying involved converting any negative exponents to their positive counterparts and combining terms where possible.

• Multiply the powers when using the power rule.
• Change negative exponents to fractions to simplify.

With these steps, complex expressions become much more manageable.

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and operations. They are a foundational part of algebra, and understanding them is crucial for solving equations and simplifying expressions.

In our example, the expression \((4x^3y^{-4})^{-2}\) is an algebraic expression that involves variables \(x\) and \(y\) raised to different powers. Simplifying it using rules of exponents transforms it into a more straightforward form.
Often, these expressions can be manipulated to find solutions or to make calculations easier, reflecting the beauty and utility of algebra.