Problem 59 - Algebra and Trigonometry

Question

\(47-72\) . Simplify the expression, and eliminate any negative exponent(s). $$ \frac{\left(x y^{2} z^{3}\right)^{4}}{\left(x^{2} y^{2} z\right)^{3}} $$

Step-by-Step Solution

Answer

\(\frac{y^{2}z^{9}}{x^{2}}\)

Step 1: Expand the Numerator

We start by expanding the numerator. Given \((xy^{2}z^{3})^{4}\), we apply the power rule \((a^m)^n = a^{mn}\) for each element in the expression.\[(xy^{2}z^{3})^{4} = x^{1 \times 4}y^{2 \times 4}z^{3 \times 4} = x^{4}y^{8}z^{12}\]

Step 2: Expand the Denominator

Next, we expand the denominator. Applying the power rule to each element in \((x^2y^2z)^3\), we have:\[(x^2y^2z)^{3} = x^{2 \times 3}y^{2 \times 3}z^{1 \times 3} = x^{6}y^{6}z^{3}\]

Step 3: Divide the Powers

Now, we divide the powers from the numerator by those in the denominator using the quotient rule \(a^m/a^n = a^{m-n}\) for each corresponding base.\[\frac{x^4y^8z^{12}}{x^6y^6z^3} = x^{4-6}y^{8-6}z^{12-3} = x^{-2}y^{2}z^9\]

Step 4: Eliminate the Negative Exponent

To eliminate the negative exponent, we rewrite \(x^{-2}\) as \(\frac{1}{x^2}\). Thus, the entire expression becomes:\[\frac{y^{2}z^{9}}{x^{2}}\]

Key Concepts

Power Rule

The Power Rule is a crucial part of simplifying algebraic expressions involving exponents. It states that when you raise a power to another power, you multiply the exponents. This can be expressed as

• \((a^m)^n = a^{m \times n}\)

When you see something like \((xy^{2}z^{3})^{4}\), you need to apply the power rule individually to each base:

• \(x^{1 \times 4} = x^4\)
• \(y^{2 \times 4} = y^8\)
• \(z^{3 \times 4} = z^{12}\)

This conversion makes it much easier to manage when the expression is more complex. It allows you to ensure that every part of the equation is correctly expanded before moving on to more advanced steps.

Quotient Rule

The Quotient Rule is essential for dividing powers with the same base. It is another key concept when simplifying algebraic expressions with exponents. When dividing like bases, you subtract the exponent in the denominator from the exponent in the numerator, represented as:

• \(a^m / a^n = a^{m-n}\)

In our example, for the expression \(\frac{x^4y^8z^{12}}{x^6y^6z^3}\), each base in the numerator is divided by its corresponding base in the denominator:

• \(x^{4-6} = x^{-2}\)
• \(y^{8-6} = y^2\)
• \(z^{12-3} = z^9\)

This rule is particularly helpful in breaking down complex expressions, so you can manage each piece effectively.

Negative Exponents

Handling negative exponents becomes straightforward once you understand their meaning. A negative exponent indicates the reciprocal of the base raised to the corresponding positive exponent:

• \(a^{-n} = \frac{1}{a^n}\)

In our exercise, we dealt with \(x^{-2}\). To eliminate the negative exponent, use the reciprocal rule:

• \(x^{-2} = \frac{1}{x^2}\)

This changes our expression from \(x^{-2}y^2z^9\) to \(\frac{y^2z^9}{x^2}\). Understanding negative exponents is key in simplification, as it allows you to express the terms without negative signs, simplifying further operations.

Exponentiation

Exponentiation refers to raising a number (the base) to a certain power (the exponent). It signifies repeated multiplication of the base. An expression like \((xy^{2}z^{3})^{4}\) means that each element inside the parentheses will be multiplied by itself a specified number of times, determined by the exponent:

• In \(x^4\), the base \(x\) is multiplied by itself 4 times.
• Similarly, \(y^8\) and \(z^{12}\) result from multiplying their respective bases the indicated number of times.

This process is critical in algebra, allowing us to manage large and complex expressions more easily by reducing them to simpler forms. Mastering exponentiation gives you the confidence to handle multiple, simultaneous computations that involve repeated operations.