Problem 4 - Contemporary Precalculus

Question

Simplify the expression. Assume \(a, b, c, d>0\) $$\left(\sqrt[3]{3} x^{2} y\right)\left(\sqrt[3]{9} x^{-1 / 3} y^{3 / 5}\right)^{-2}$$

Step-by-Step Solution

Answer

Question: Simplify the expression \(\sqrt[3]{3} x^2 y (\sqrt[3]{9} x^{-1/3} y^{3/5})^{-2}\), assuming \(a, b, c, d > 0\). Answer: \(x^{\frac{8}{3}}y^{-\frac{1}{5}}\)

Step 1: Simplify each factor individually

First, we simplify the first factor, \(\sqrt[3]{3} x^2 y\). There's not much to do here since the terms are already in their simplest form. Now, we simplify the second factor, \(\sqrt[3]{9} x^{-1/3} y^{3/5}\). The cube root of 9 can be written as \(9^{1/3}\). So the second factor becomes: $$9^{1/3} x^{-1/3} y^{3/5}$$

Step 2: Simplify the expression raised to the power of -2

Next, we need to apply the power -2 to the second factor which is: $$\left(9^{1/3} x^{-1/3} y^{3/5}\right)^{-2}$$ Applying the power -2 to each term, we have: $$\left(9^{-2/3}\right)\left(x^{\frac{2}{3}}\right)\left(y^{-\frac{6}{5}}\right)$$

Step 3: Combine both factors

Now, we need to multiply the first factor, which is \(\sqrt[3]{3} x^2 y\), by the second factor that we just simplified, which is \(\left(9^{-2/3}\right)\left(x^{\frac{2}{3}}\right)\left(y^{-\frac{6}{5}}\right)\). Multiplying the terms together, we get: $$\left(\sqrt[3]{3} x^2 y\right)\left(\left(9^{-2/3}\right)\left(x^{\frac{2}{3}}\right)\left(y^{-\frac{6}{5}}\right)\right)$$ $$=\left(\sqrt[3]{3}\cdot9^{-2/3}\right)\left(x^{2+\frac{2}{3}}\right)\left(y^{1-\frac{6}{5}}\right)$$

Step 4: Simplify the expression

Now, we simplify the final expression: $$=\left(3^{\frac{1}{3}}\cdot9^{-\frac{2}{3}}\right)x^{\frac{8}{3}}y^{-\frac{1}{5}}$$ Since both \(3^{\frac{1}{3}}\) and \(9^{-\frac{2}{3}}\) have the same base, we can rewrite \(9^{-\frac{2}{3}}\) as \((3^2)^{-\frac{2}{3}}\) and then combine them: $$=\left(3^{\frac{1}{3}}\cdot(3^2)^{-\frac{2}{3}}\right)x^{\frac{8}{3}}y^{-\frac{1}{5}}$$ $$=3^{\frac{1}{3}-\frac{4}{3}}x^{\frac{8}{3}}y^{-\frac{1}{5}}$$ $$=3^{-\frac{3}{3}}x^{\frac{8}{3}}y^{-\frac{1}{5}}$$ $$=x^{\frac{8}{3}}y^{-\frac{1}{5}}$$ The simplified expression is: $$x^{\frac{8}{3}}y^{-\frac{1}{5}}$$

Key Concepts

Simplifying Expressions

Simplifying expressions involves reducing them to their simplest form without changing their value. This often includes performing operations like factoring, combining like terms, and using known mathematical properties, such as the distributive or associative properties. In our exercise, the expression is made up of multiple terms that involve various operations like multiplication and exponentiation.

One crucial step is recognizing whether any parts of the expression can be simplified individually before attempting to simplify the entire expression. For example, the cube root of a number or raising a term to a power (like \(-2\)) can often be simplified separately.

• Factorize numbers and expressions by finding common elements.
• Apply basic arithmetic rules to combine like terms.
• Simplify powers and roots systematically by using exponent rules.

Breaking an expression down into manageable parts, and dealing with each part step by step, helps in not only understanding the content better but also in arriving at a correct simplified form.

Negative Exponents

Negative exponents can seem tricky at first, but they simply represent the reciprocal of a number raised to a positive exponent. For instance, if you encounter something like \(x^{-n}\), it means \(\frac{1}{x^n}\).

In our exercise, raising an expression to the power of \(-2\) involves multiplying the reciprocal of each factor. This is how \((9^{1/3} x^{-1/3} y^{3/5})^{-2}\) becomes \(9^{-2/3} x^{2/3} y^{-6/5}\):

• First, apply the negative exponent to each factor separately.
• Invert the factor and then raise it to the positive power.
• It's important to handle these transformations carefully to maintain the integrity of the expression.

By mastering the manipulation of negative exponents, you can significantly simplify many expressions, reducing them into much more manageable components.

Fractional Exponents

Fractional exponents express roots in a compact form. For instance, a square root of a number can be written as \( a^{1/2} \), and a cube root as \( a^{1/3} \). This notation generalizes to any root.

Dealing with fractional exponents follows similar rules to those of integer exponents but often involves simplification through the understanding of roots. Let's see what happens when we combine two numbers with fractional exponents:

• Fractional exponents work as roots, akin to calculating the cube root of \(9\) as \(9^{1/3}\).
• Apply the rule: \( (a^m)^n = a^{m \cdot n} \) when dealing with expressions like raising to a power involving fractions.

In our example, simplifying \( \sqrt[3]{3} \cdot 9^{-2/3} \) requires converting the numbers into fractional exponents and applying the combination rules. Understanding fractional exponents enhances your ability to deal with complex expressions efficiently.