Problem 68 - Algebra 1: Concepts and Skills

Question

Simplify the expression \(\frac{x^{-2}}{y^{-3}} \cdot\left(\frac{x}{y}\right)^{-1}\) F. \(\frac{y^{2}}{x}\) G. \(\frac{y^{4}}{x^{3}}\) H. \(\frac{x^{2}}{y^{3}}\) J. \(\frac{x^{3}}{y^{4}}\)

Step-by-Step Solution

Answer

The simplified expression is \(\frac{y^{4}}{x^{3}}\), hence the answer is G

Step 1: Deal with the negative exponent

Let's first rewrite the expression by applying the rule that \(a^{-n} = 1/a^n\). Thus, we can write \(x^{-2}\) as \(\frac{1}{x^2}\) and \(y^{-3}\) as \(\frac{1}{y^3}\). This changes the expression into: \[\frac{\frac{1}{x^{2}}}{\frac{1}{y^{3}}}\cdot\left(\frac{x}{y}\right)^{-1}\]

Step 2: Simplify the fraction

Here, we can use the rule that \(\frac{1/a}{1/b} = b/a\) to simplify the expression further. This changes the expression into: \[\frac{y^{3}}{x^{2}}\cdot\left(\frac{x}{y}\right)^{-1}\]

Step 3: Deal with the exponent in the parenthesis

For the expression \((x/y)^{-1}\), we can use the rule \((a/b)^{-n} = (b/a)^n\) to rewrite it as \((y/x)^1\), or simply \(y/x\). So now the expression becomes: \[\frac{y^{3}}{x^{2}}\cdot\frac{y}{x}\]

Step 4: Multiply the terms

Now we multiply the terms: \[y^{3} \cdot y = y^{4}\] \[x^{2} \cdot x = x^{3}\] So the expression simplifies to: \[\frac{y^{4}}{x^{3}}\]

Key Concepts

Negative Exponents

Negative exponents might seem tricky at first, but they follow a simple rule. When you have a negative exponent, you are essentially taking the reciprocal of the base raised to the positive of that exponent. For instance, if you have \(x^{-2}\), it means \(\frac{1}{x^{2}}\). This is because the negative exponent indicates how many times you have to divide by the base instead of multiply.

Here are a few examples to clarify:

• \(a^{-1} = \frac{1}{a}\)
• \(b^{-3} = \frac{1}{b^{3}}\)
• \(\left(\frac{2}{3}\right)^{-2} = \left(\frac{3}{2}\right)^2 = \frac{9}{4}\)

When you see a negative exponent, think about flipping the fraction. It makes life easier when simplifying expressions involving negative powers.

Exponent Rules

Understanding exponent rules can greatly simplify expressions, especially those containing negative or fractional exponents. The power rules are essential, and they include:

• Product of Powers: When multiplying with the same base, add the exponents: \(a^m \cdot a^n = a^{m+n}\).
• Quotient of Powers: When dividing with the same base, subtract the exponents: \(\frac{a^m}{a^n} = a^{m-n}\).
• Power of a Power: Raise one exponent to another by multiplying: \((a^m)^n = a^{m \times n}\).

Using these rules, you can transform complex expressions into simpler ones efficiently. In our example, we reduced complexity by applying the rules to both negative exponents and the fractions involved in the expression.

Fractional Exponents

Fractional exponents represent roots, making them an alternative way to write radicals. The exponent \(\frac{1}{2}\) stands for the square root, \(\frac{1}{3}\) for the cube root, and so on. Here's how they work:

• \(a^{\frac{1}{n}} = \sqrt[n]{a}\)
• \(a^{\frac{m}{n}} = \sqrt[n]{a^m} = (a^m)^{\frac{1}{n}}\)

For example, \(x^{\frac{3}{2}}\) means the same as \(\sqrt{x^3}\), or \((\sqrt{x})^3\). Understanding these connections allows you to easily transform and solve expressions involving exponents.

If an expression contains both negative and fractional exponents, apply the rules separately for clarity, then combine the results in one step by step simplification. This structured approach helps to prevent errors and enhances understanding.