Problem 49 - Algebra for College Students

Question

Simplify each expression. Express final results without using zero or negative integers as exponents. \(\left(x^{2} y^{-6}\right)^{-1}\)

Step-by-Step Solution

Answer

The expression simplifies to \( \frac{y^{6}}{x^{2}} \).

Step 1: Apply the Power of a Power Rule

The expression is \( \left(x^{2} y^{-6}\right)^{-1} \). We need to distribute the outside exponent \(-1\) to each term inside the parenthesis by applying the power of a power rule, \( (a^m)^n = a^{m \cdot n} \). So, \( x^{2} \) raised to \(-1\) becomes \( x^{-2} \) and \( y^{-6} \) raised to \(-1\) becomes \( y^{6} \). The expression simplifies to \( x^{-2} y^{6} \).

Step 2: Convert Negative Exponents to Positive

To express the result without negative exponents, we need to move the terms with negative exponents to the denominator. Since \( x^{-2} = \frac{1}{x^{2}} \), the expression becomes \( \frac{y^{6}}{x^{2}} \).

Step 3: Final Simplified Expression

The final expression with only positive exponents and no zero exponents is \( \frac{y^{6}}{x^{2}} \).

Key Concepts

Negative Exponents

When you see an expression with negative exponents, like \( x^{-2} \), it might seem a bit daunting at first. But worry not, it's simpler than it appears! A negative exponent means that the base is on the wrong side of the fraction line. To convert a negative exponent into a positive one, just take the reciprocal of the base. In easier terms, think of \( x^{-2} \) as \( \frac{1}{x^2} \).

Here's a quick refresher of how to deal with negative exponents:

• Move the base with the negative exponent to the opposite side of the fraction line.
• Once it's on the opposite side, the exponent turns positive.
• Keep practicing with different numbers to master this concept effortlessly.

With practice, you'll find that simplifying expressions with negative exponents becomes second nature.

Power of a Power Rule

The power of a power rule is essential when dealing with expressions like \( \left(a^m\right)^n \). It tells us that you can multiply the exponents together to simplify the expression: \( a^{m \cdot n} \). This is incredibly useful, as it turns expressions that seem difficult into something much easier to handle.

For example, consider \( \left(x^2 y^{-6}\right)^{-1} \), the first step is to apply this rule. You would multiply each exponent inside the parentheses by the exponent outside the parentheses, which in this case is \(-1\). So, you get \( x^{-2} \) and \( y^{6} \).

Remember:

• Multiply the exponents when you have a base raised to a power, which is then raised to another power.
• This rule helps in breaking down complex expressions into manageable ones.

With this rule in your toolkit, simplifying complicated expressions becomes much more intuitive.

Algebraic Fractions

Algebraic fractions might look a bit intimidating, but they follow the same rules as regular fractions. The key is to simplify them by making sure there are no negative exponents remaining. In the expression \( \frac{y^6}{x^2} \), we have taken an algebraic fraction originating from \( x^{-2} y^6 \).

Here's what you should remember about algebraic fractions:

• Make sure the expression is fully simplified by getting rid of any negative exponents.
• Shift terms with negative exponents into the denominator to ensure all exponents are positive.
• Use basic fraction operations like multiplication and division to manage algebraic fractions just as you would with numerical fractions.

By consistently applying these strategies, managing and simplifying algebraic fractions will become a skill you can count on in any mathematical problem.