Problem 38 - College Algebra

Question

For the following exercises, simplify the given expression. Write answers with positive exponents. $$ \frac{(16 \sqrt{x})^{2}}{y^{-1}} $$

Step-by-Step Solution

Answer

The simplified expression is \(256xy\).

Step 1: Apply Power to Each Component

The given expression is \( \frac{(16 \sqrt{x})^2}{y^{-1}} \). Start by applying the power of 2 to each component of the numerator inside the parenthesis, which is \((16 \sqrt{x})^2\). This means you need to square both 16 and \(\sqrt{x}\): - 16 squared is 256.- \((\sqrt{x})^2\) is just \(x\). Therefore, the expression becomes \( \frac{256x}{y^{-1}} \).

Step 2: Simplify to Positive Exponents

The expression is \( \frac{256x}{y^{-1}} \). To simplify this and to express it with positive exponents, remember that \(y^{-1}\) is equivalent to \(\frac{1}{y}\). Thus, dividing by \(y^{-1}\) is equivalent to multiplying by \(y\). Hence, the expression becomes:\( 256xy \).

Key Concepts

Power of a Power

When simplifying expressions like \((16 \sqrt{x})^{2}\), you encounter the **Power of a Power** rule. This concept states that when you raise a power to another power, you multiply the exponents. For example, if you have \((a^m)^n\), it simplifies to \(a^{m \cdot n}\).

In the exercise, \((16 \sqrt{x})^{2}\) involves squaring both 16 and the square root of \(x\). Calculating 16 squared gives 256, because 16 multiplied by itself equals 256. For \((\sqrt{x})^2\), it simplifies directly to \(x^1\) or just \(x\), because the square and the square root cancel each other out. This technique is useful for tackling complex expressions where multiple powers are involved.

Positive Exponents

A key part of simplifying expressions is ensuring all exponents are positive. **Positive Exponents** make an expression clearer and easier to interpret. Any negative exponent can be converted to a positive one by taking the reciprocal.

In the example \(\frac{256x}{y^{-1}}\), converting \(y^{-1}\) involves writing it as a reciprocal: \(y^{-1} = \frac{1}{y}\). After simplifying, the expression changes to \(256 \cdot x \cdot y\), eliminating any negative exponents. This step ensures the expression not only looks cleaner but also adheres to standard algebraic format.

Square Roots

**Square Roots** are inverse operations to squaring a number. When you see a square root, like \(\sqrt{x}\), it implies finding a number that, when squared, gives \(x\).

Upon squaring \(\sqrt{x}\), as seen in the expression \((\sqrt{x})^2\), you essentially cancel out the square root, resulting in \(x\). This property is pivotal when simplifying terms involving both square roots and exponents, as it helps to reduce the complexity of the expression. Understanding this relationship makes it easier to navigate problems where powers and radicals intersect.

Division of Exponents

Understanding the **Division of Exponents** involves knowing how to handle terms with exponents within a fraction. This principle can be simplified by subtracting the exponents of the numerator and the denominator if they share the same base.

However, in the given exercise, the focus was more on converting \(y^{-1}\) to positive using its equivalent fraction \(\frac{1}{y}\). When dividing by \(y^{-1}\), it results in \(y\) moving to the numerator, effectively making it a multiplication operation instead of a division. This clever technique helps streamline expressions quickly and is a powerful tool for simplifying exponent-based expressions.