Problem 46 - Algebra 2

Question

Write each expression in simplest form. Assume that all variables are positive. $$\left(x^{\frac{1}{2}} y^{-\frac{2}{3}}\right)^{-6}$$

Step-by-Step Solution

Answer

The simplified form of the expression \(\left(x^{\frac{1}{2}} y^{-\frac{2}{3}}\right)^{-6}\) is \(\frac{1}{x^{3} y^{4}}\).

Step 1: Apply the Power of a Power rule

The exercise involves simplifying an expression with a negative exponent. Begin by applying the power of a power rule to the expression. As a result, \(\left(x^{\frac{1}{2}} y^{-\frac{2}{3}}\right)^{-6}\) becomes \(x^{-6*\frac{1}{2}} y^{6*-\frac{2}{3}}\).

Step 2: Simplify the Expression

We now simplify the new expression: \(x^{-6*\frac{1}{2}} y^{6*-\frac{2}{3}}\) simplifies to \(x^{-3} y^{-4}\).

Step 3: Apply the Negative Exponent Rule

We now apply the rule for negative exponents: \(a^{-m} = 1/a^{m}\). Therefore, \(x^{-3} y^{-4}\) becomes \(\frac{1}{x^{3}} \frac{1}{y^{4}}\).

Step 4: Combine the Fractions

Finally, we merge the fractions to have one simplified expression. The expression \(\frac{1}{x^{3}} \frac{1}{y^{4}}\) simplifies to \(\frac{1}{x^{3} y^{4}}\).

Key Concepts

Power of a Power Rule

The "power of a power" rule is a handy tool in simplifying expressions that involve exponents. When you see an expression raised to another power, this rule helps in simplifying by multiplying the exponents. For an expression like \((a^{m})^{n}\), according to the power of a power rule, you multiply the exponents: \(a^{m \cdot n}\).

In our exercise, the expression \((x^{\frac{1}{2}} y^{-\frac{2}{3}})^{-6}\) is simplified using this rule.

• For the variable \(x\), the exponent is multiplied: \(\frac{1}{2} \times -6 = -3\). So, \(x^{\frac{1}{2}}\) becomes \(x^{-3}\).
• For the variable \(y\), a similar operation occurs: \(-\frac{2}{3} \times -6 = 4\). Thus, \(y^{-\frac{2}{3}}\) transforms into \(y^{-4}\).

Using the power of a power rule can seem puzzling at first, but remember that scaling exponents is its main function. It's a straightforward multiplication task that simplifies potentially complex expressions.

Negative Exponent Rule

Negative exponents are another essential concept in algebra. They can seem confusing but remember: a negative exponent simply means you take the reciprocal of the base raised to the opposite exponent. Using the rule \(a^{-m} = \frac{1}{a^{m}}\), you turn a negative exponent into a positive one by flipping the base into a fraction.

In the provided exercise, we see both variables \(x\) and \(y\) with negative exponents.

• \(x^{-3}\) transforms to \(\frac{1}{x^{3}}\)
• \(y^{-4}\) turns into \(\frac{1}{y^{4}}\)

This step is crucial as it allows expressions to be presented without negative exponents, making them simpler and friendlier for further operations or interpretations. Utilizing the negative exponent rule clarifies the expression and aids in converting it into standard fraction form.

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and operations. Understanding how to manipulate them through various rules and operations is vital in algebra.

For example, in the exercise, we dealt with a complex expression initially. Simplifying it through structured rules reveals the fundamental nature of algebraic work.

Here are some key insights:

• Algebraic expressions may have variables with exponents, indicating repeated multiplication, which sometimes need special rules for simplification.
• Combining and re-writing these expressions using rules like the power of a power and handling negative exponents are part of simplifying processes in algebra.
• Ultimately, simplification helps make complex expressions easier to evaluate and understand, transforming something like \((x^{\frac{1}{2}} y^{-\frac{2}{3}})^{-6}\) into a simple fraction \(\frac{1}{x^{3}y^{4}}\).

By really getting to know algebraic expressions and their characteristics, you set a solid foundation for problem-solving in more advanced mathematics.