Problem 68 - Algebra A Combined Function

Question

Simplify each expression. Write each result using positive exponents only. $$ \frac{\left(-3 x^{2} y^{2}\right)^{-2}}{(x y z)^{-2}} $$

Step-by-Step Solution

Answer

\( \frac{z^{2}}{9x^{2}y^{2}} \) with positive exponents only.

Step 1: Apply the Negative Exponent Property

Recall that the expression with a negative exponent, such as \( a^{-n} \), can be rewritten as \( \frac{1}{a^{n}} \) and vice versa. Apply this property to each part of the expression \( \left(-3x^{2}y^{2}\right)^{-2} \) and \( (xy z)^{-2} \). We rewrite: \[ \frac{1}{((-3x^{2}y^{2})^{2})} \div \frac{1}{((xyz)^{2})}. \]

Step 2: Simplify the Complex Fraction

Simplify the complex fraction using the property \( \frac{\frac{a}{b}}{\frac{c}{d}} = \frac{a}{b} \times \frac{d}{c} \). So, simplify the expression: \[ \frac{1}{(-3x^{2}y^{2})^{2}} \times (xyz)^{2}. \] This becomes \[ \frac{(xyz)^{2}}{(-3x^{2}y^{2})^{2}}. \]

Step 3: Expand Squared Terms

Expand both the numerator and the denominator by squaring each term:\[ (xyz)^{2} = x^{2}y^{2}z^{2}. \] For the denominator: \[ (-3x^{2}y^{2})^{2} = (-3)^{2} \cdot (x^{2})^{2} \cdot (y^{2})^{2} = 9x^{4}y^{4}. \]

Step 4: Divide Each Corresponding Term

Now, divide each corresponding term from the numerator by the denominator:\[ \frac{x^{2}y^{2}z^{2}}{9x^{4}y^{4}} = \frac{x^{2}}{x^{4}} \cdot \frac{y^{2}}{y^{4}} \cdot \frac{z^{2}}{9}. \]Use the property of exponents, \( \frac{a^{m}}{a^{n}} = a^{m-n} \), to simplify further.

Step 5: Simplify Using Exponents Property

Applying the property:\[ \frac{x^{2}}{x^{4}} = x^{2-4} = x^{-2}, \]\[ \frac{y^{2}}{y^{4}} = y^{2-4} = y^{-2}, \]So the expression simplifies to:\[ \frac{x^{-2}y^{-2}z^{2}}{9}. \]

Step 6: Convert to Positive Exponents

Convert the negative exponents to positive by rewriting them using the reciprocal property:\[ x^{-2} = \frac{1}{x^{2}}, \quad y^{-2} = \frac{1}{y^{2}}. \]Combine these with \( z^{2} \) to get:\[ \frac{z^{2}}{9x^{2}y^{2}}. \]

Key Concepts

Negative Exponents

Negative exponents might seem tricky at first, but they actually make our calculations easier. A negative exponent, such as \( a^{-n} \), signifies that we need to take the reciprocal of the base. Instead of multiplying \( a \) by itself \( n \) times, you're essentially dividing one by \( a^n \). For example, if you have \( 2^{-3} \), it translates to \( \frac{1}{2^3} \), which equals \( \frac{1}{8} \).
This rule is pivotal when simplifying expressions with negative exponents because it converts these tricky-looking terms into more manageable forms.
Negative exponents help us transform complex fractions and make our answers cleaner by converting every term into a positive exponent form.

Simplifying Expressions

Simplifying expressions is the process of writing a mathematical expression as simply as possible. Involves combining like terms and reducing fractions. When tackling expressions with exponents, the aim is to reduce the expression to a form that is easy to handle and understand. Take, for instance, the expression \[ \frac{(xyz)^2}{(-3x^2y^2)^2} \]. To simplify this, start by squaring each term in the numerator and the denominator. Doing this step results in \( x^2y^2z^2 \) in the numerator and \( 9x^4y^4 \) in the denominator.
After expanding each portion, the goal is to

• divide like terms,
• use exponent rules,
• reduce everything to its simplest positive form.

Simplifying isn't just about getting a smaller expression; it's about understanding and manipulating the algebraic structure.

Exponent Rules

Exponent rules serve as shortcuts that make working with powers straightforward. When dealing with powers, these rules guide us in adding, subtracting, and multiplying exponents accurately. Here are a few essential rules:

• The Product of Powers Rule states \( a^m \times a^n = a^{m+n} \).
• The Quotient of Powers Rule expresses \( \frac{a^m}{a^n} = a^{m-n} \).
• The Power of a Power Rule gives \( (a^m)^n = a^{m\times n} \).
• The Zero Exponent Rule tells us \( a^0 = 1 \) for any \( a eq 0 \).

When simplifying \( \frac{x^2y^2z^2}{9x^4y^4} \), these rules are extremely helpful. By applying the Quotient of Powers Rule, we simplify it to \( x^{-2}y^{-2}z^2 \), and then use the property of converting negative exponents to reformat it as \( \frac{z^2}{9x^2y^2} \) with only positive exponents. Memorizing these rules can provide clarity and speed up the process of simplifying mathematical expressions involving exponents.