Problem 67 - Algebra and Trigonometry

Question

\(49-68=\) Simplify the expression and eliminate any negative exponent(s). Assume that all letters denote positive numbers. $$ \left(\frac{a^{1 / 6} b^{-3}}{x^{-1} y}\right)^{3}\left(\frac{x^{-2} b^{-1}}{a^{3 / 2} y^{1 / 3}}\right) $$

Step-by-Step Solution

Answer

The simplified expression is \(\frac{x}{a b^{10} y^{10/3}}\).

Step 1: Simplify the First Fraction Inside the Brackets

Simplify the first fraction \(\left(\frac{a^{1 / 6} b^{-3}}{x^{-1} y}\right)\). Apply the property of exponents, \(\frac{x^m}{x^n} = x^{m-n}\), so you get \(a^{1/6}b^{-3}x^1y^{-1}\).

Step 2: Simplify the Second Fraction Inside the Brackets

Simplify the second fraction \(\left(\frac{x^{-2} b^{-1}}{a^{3/2} y^{1/3}}\right)\) using the same property of exponents to get \(x^{-2}b^{-1}a^{-3/2}y^{-1/3}\).

Step 3: Combine the Expressions

Combine the simplified versions from Step 1 and Step 2: \( \left(a^{1/6}b^{-3}x^1y^{-1}\right)^3 \cdot \left(x^{-2}b^{-1}a^{-3/2}y^{-1/3}\right) \).

Step 4: Apply the Power to the First Term

Distribute the exponent 3 to each term in \((a^{1/6}b^{-3}x^1y^{-1})^3\). This gives \(a^{1/2}b^{-9}x^3y^{-3}\).

Step 5: Combine All Terms

Multiply all terms: \(a^{(1/2)}a^{-3/2}b^{-9}b^{-1}x^{3}x^{-2}y^{-3}y^{-1/3}\). Simplify by adding exponents: \(a^{-1}b^{-10}x^1y^{-10/3}\).

Step 6: Final Simplification

Rewrite the expression with positive exponents: \(\frac{x}{a b^{10} y^{10/3}}\).

Key Concepts

Exponent Rules

When working with expressions involving exponents, understanding the rules governing their manipulation is crucial. Exponent rules help us simplify expressions and make calculations easier. These rules are applicable when multiplying, dividing, and raising exponents to powers.

Here are some key exponent rules:

• Product of Powers: When multiplying like bases, add the exponents, i.e., \(x^m \cdot x^n = x^{m+n}\).
• Quotient of Powers: For like bases, subtract the exponent of the denominator from the exponent of the numerator, i.e., \(\frac{x^m}{x^n} = x^{m-n}\).
• Power of a Power: Raise a power to another power by multiplying the exponents, i.e., \((x^m)^n = x^{m \cdot n}\).
• Power of a Product: Distribute the exponent to each factor within the parentheses, i.e., \((xy)^n = x^n y^n\).
• Power of a Quotient: Distribute the exponent to both the numerator and the denominator, i.e., \(\left(\frac{x}{y}\right)^n = \frac{x^n}{y^n}\).

Understanding these rules will enable you to handle expressions like those in the exercise, simplifying complex terms more efficiently.

Negative Exponents

Negative exponents can initially seem a bit tricky, but they play an important role in making calculations easier. A negative exponent indicates the reciprocal of the base raised to the opposite positive exponent.

Here's what you need to know about negative exponents:

• A base with a negative exponent, \(x^{-n}\), is equivalent to \(\frac{1}{x^n}\).
• Conversely, \(\frac{1}{x^{-n}}\) simplifies to \(x^n\).

Understanding negative exponents allows you to rewrite parts of an expression from the example like \(b^{-3}\) as \(\frac{1}{b^3}\). In our exercise, this helps in rearranging terms to eliminate the negative exponent if needed, ultimately making the expression simpler to work with.

Fractional Exponents

Fractional exponents represent roots and are a compact and elegant way of expressing them. They help us transform radicals into exponents, simplifying the calculations further.

Key points on fractional exponents include:

• \(x^{1/n}\) is the same as the \(n^{th}\) root of \(x\), i.e., \(x^{1/n} = \sqrt[n]{x}\).
• If you come across \(x^{m/n}\), it's equivalent to the \(n^{th}\) root of \(x^m\), i.e., \(x^{m/n} = (\sqrt[n]{x})^m\).
• Fractional exponents abide by the same rules as integer exponents, for multiplication, division, and powers.

In our exercise scenario, expressions such as \(a^{1/6}\) and \(a^{3/2}\) are handled using these principles. By converting root expressions to fractional exponents, simplification becomes a matter of applying exponent rules more directly.