Problem 49 - Algebra A Combined Function

Question

Use the power rule and the power of a product or quotient rule to simplify each expression. $$ \left(\frac{-2 x z}{y^{5}}\right)^{2} $$

Step-by-Step Solution

Answer

\( \frac{4x^2z^2}{y^{10}} \)

Step 1: Apply the Power to Each Component

The expression \( \left( \frac{-2 x z}{y^{5}} \right)^{2} \) requires us to apply the square to both the numerator and the denominator independently. Split this as follows:\[ \left(\frac{-2xz}{y^5}\right)^2 = \frac{(-2xz)^2}{(y^5)^2} \]

Step 2: Apply the Power to the Numerator

Now, focus on \((-2 x z)^2\). Use the power of a product rule to apply the exponent to each term inside the parentheses:\[ (-2 x z)^2 = (-2)^2 \cdot x^2 \cdot z^2 \]Calculate \((-2)^2\):\((-2)^2 = 4\),So, the numerator becomes:\[ 4x^2z^2 \]

Step 3: Apply the Power to the Denominator

For the denominator \((y^5)^2\), use the power rule \((a^m)^n = a^{m \cdot n}\):\[ (y^5)^2 = y^{5 \cdot 2} = y^{10} \]

Step 4: Combine Numerator and Denominator

Now, combine the results from Steps 2 and 3 to obtain the simplified expression:\[ \frac{4x^2z^2}{y^{10}} \]

Key Concepts

Exponents

Exponents are a mathematical notation indicating the number of times a number, known as the base, is multiplied by itself. In the expression \((-2xz)^2\), \(2\) is the exponent and \((-2xz)\) is the base. This tells us to multiply \(-2xz\) by itself.

• The power of a product rule allows you to apply the exponent to each factor within the parentheses. For example, in \((-2xz)^2\), the exponent \(2\) is applied separately to \(-2\), \(x\), and \(z\).
• Calculating \((-2)^2\) gives \(4\), because \((-2) imes (-2) = 4\).
• Incorporate the exponent onto \(x\) and \(z\) to give \(x^2\) and \(z^2\).

Remember that negative exponents imply division and that zero as an exponent results in \(1\). Understanding how to manipulate exponents is crucial for simplifying algebraic expressions.

Numerator and Denominator

The terms "numerator" and "denominator" are crucial for understanding fractions and expressions involving quotients. They define the two parts of a fraction; the numerator is the top portion, while the denominator sits below the fraction line.

• The numerator tells us how many parts we have, while the denominator shows the number of equal parts the whole is divided into.
• In this expression \(\frac{-2xz}{y^5}\), \(-2xz\) is the numerator and \(y^5\) is the denominator.

When we apply a power, such as for \(\left(\frac{-2xz}{y^5}\right)^2\), the exponent needs to be applied separately to both the numerator and the denominator. This leads to \((-2xz)^2\) for the numerator and \((y^5)^2\) for the denominator.
This method ensures each part of the fraction is correctly addressed in simplification, maintaining the integrity of the expression.

Simplifying Expressions

Simplifying expressions means reducing them to their most basic form while maintaining the same value. This often involves using the rules for exponents and understanding how to handle fractions correctly. Simplifying helps in solving equations more easily or making expressions more readable.

• Use the power rule: \((a^m)^n = a^{m \cdot n}\). This shows how, for instance, \((y^5)^2 = y^{10}\).
• Simplify the multiplication of numbers directly: for example, \((-2)^2 = 4\).

Combine the simplified numerator \(4x^2z^2\) and the denominator \(y^{10}\) into a single fraction: \(\frac{4x^2z^2}{y^{10}}\).
Being fluent in simplifying expressions will make more complicated algebraic operations manageable and is a fundamental skill for any math student.