Problem 48 - Algebra A Combined Function

Question

Simplify each expression. Write each result using positive exponents only. $$ \left(5 x^{3} y^{-2}\right)^{2} $$

Step-by-Step Solution

Answer

The simplified expression is \( \frac{25x^6}{y^4} \).

Step 1: Apply the Power Rule to Each Part of the Expression

To simplify an expression with exponents, we can use the power rule, which states that \[ (a^m)^n = a^{m imes n} \]Applying this to the expression, we have:\[ \ (5x^3y^{-2})^2 = 5^2(x^3)^2(y^{-2})^2 \]

Step 2: Simplify Each Component

Now, simplify each component separately using the power rule:1. For the coefficient: - \(5^2 = 25\)2. For the \(x\) term: - \((x^3)^2 = x^{3 imes 2} = x^6\)3. For the \(y\) term: - \((y^{-2})^2 = y^{-2 imes 2} = y^{-4}\)

Step 3: Write the Final Expression with Positive Exponents

To express the result with positive exponents for all terms, recall that a negative exponent represents a reciprocal:- \(y^{-4} = \frac{1}{y^4}\)Thus, the final expression is:\[ 25x^6 \cdot \frac{1}{y^4} = \frac{25x^6}{y^4} \]

Key Concepts

Power Rule in Exponents

Understanding the power rule in exponents is crucial when dealing with expressions that include powers raised to other powers. This rule states:

• \((a^m)^n = a^{m \times n}\)

If you have a base 'a', raised to the power 'm', and that quantity is again raised to another power 'n', you multiply the exponents 'm' and 'n'.

Let’s apply this concept to the expression \((5x^3y^{-2})^2\). Each part of the expression, including the coefficient and the variables, is raised to the power 2. Breaking it down:

• The coefficient is \(5\) which becomes \(5^2\).
• The \(x\) term \(x^3\) becomes \((x^3)^2 = x^{3 \times 2} = x^6\).
• Similarly, the \(y\) term \(y^{-2}\) becomes \((y^{-2})^2 = y^{-2 \times 2} = y^{-4}\).

This initial step sets the foundation for simplifying the expression further.

Simplifying Expressions

Once the power rule is applied, the next task is simplifying the expression. Simplification aims to express the whole term in its most elementary form. Here’s how each component is simplified:

• For the coefficient: Calculate \(5^2\), giving you 25.
• For the \(x\) term: Calculate \((x^3)^2\) to get \(x^6\).
• For the \(y\) term: Calculate \((y^{-2})^2\) which results in \(y^{-4}\).

After simplifying each component separately, they are combined to form the expression:
\[25x^6y^{-4}\]
Simplifying expressions is like clearing the fog around mathematical terms to get a clearer view of what they represent. Each step must be precise, ensuring the expression retains its integrity while being easier to understand.

Positive Exponents

Expressions are often required to be written using positive exponents, as these are simpler and more intuitive. A negative exponent implies the reciprocal of the number. Therefore, turning negative exponents into positive ones helps simplify the readability of the expression.

• A basic rule to remember: \(a^{-n} = \frac{1}{a^n}\).

In the original expression, the term \(y^{-4}\) is simplified with a positive exponent by taking the reciprocal:\[y^{-4} = \frac{1}{y^4}\]

This means our simplified expression \(25x^6y^{-4}\) can be rewritten as:
\[\frac{25x^6}{y^4}\]
Writing expressions with positive exponents prepares them for further mathematical operations and analysis, thus making the expressions easier to handle in future calculations.