Problem 41 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 11-46: Simplify. $$ \left(\frac{-8 x^{3}}{y^{-6}}\right)^{2 / 3} $$

Step-by-Step Solution

Answer

The simplified expression is \( 4x^2y^4 \).

Step 1: Understand the Expression

We are given the expression \( \left(\frac{-8 x^{3}}{y^{-6}}\right)^{2/3} \). Our goal is to simplify this expression by applying the rules of exponents.

Step 2: Simplify the Inner Fraction

Notice that the denominator \(y^{-6}\) is equivalent to \( \frac{1}{y^{6}} \). Therefore, we rewrite the expression as \( -8x^3 \cdot y^6 \). The new expression becomes \( (-8x^3y^6) \).

Step 3: Apply the Power to the Fraction

Apply the power of \( \frac{2}{3} \) to the entire expression \( (-8x^3y^6) \). This involves distributing the power of \( \frac{2}{3} \) to each term: \((-8)^{2/3} \), \((x^3)^{2/3} \), and \((y^6)^{2/3} \).

Step 4: Simplify Each Term with the Power

Calculate each term separately: - \((-8)^{2/3}\) is the cube root of \(-8\) squared. The cube root of \(-8\) is \(-2\) and \((-2)^2 = 4\).- \((x^3)^{2/3}\) means \((x^{3})^{2/3} = x^{2}\).- \((y^6)^{2/3}\) is \(y^{4}\) because 6 multiplied by \(\frac{2}{3}\) equals 4.

Step 5: Combine Simplified Components

Combine the simplified results to achieve the expression: \( 4x^2y^4 \).

Key Concepts

Exponent Rules

Exponent rules are key to simplifying expressions involving powers. By understanding these rules, one can navigate through complex expressions with ease. Common exponent rules include:

• Product Rule: This states that when multiplying two powers with the same base, you add the exponents. For instance, \( a^m imes a^n = a^{m+n} \).
• Power of a Power Rule: When raising a power to another power, multiply the exponents, like \( (a^m)^n = a^{m imes n} \).
• Quotient Rule: When dividing powers with the same base, subtract the exponents, \( \frac{a^m}{a^n} = a^{m-n} \).
• Negative Exponent Rule: A negative exponent indicates a reciprocal, so \( a^{-m} = \frac{1}{a^m} \).

Understanding these rules helps in simplifying the expression \( \left(\frac{-8 x^{3}}{y^{-6}}\right)^{2 / 3} \) efficiently. Notice how the negative exponent in the denominator was handled using the Negative Exponent Rule.

Radical Expressions

Radical expressions involve roots, such as square roots or cube roots. They appear frequently in algebra and are essential for simplifying expressions. A radical like \( \sqrt{a} \) is a number that, when squared, gives \( a \). Similarly, cube roots, denoted by \( \sqrt[3]{a} \), are numbers that, when cubed, result in \( a \).
In the exercise, we encounter the term \((-8)^{2/3}\). This involves finding the cube root of \(-8\), which results in \(-2\), as \(-2 \times -2 \times -2 = -8\).

• Once the cube root is found, apply the exponent: \((-2)^2 = 4\) as per the Power of a Power Rule.

Being comfortable with radical expressions can help in understanding fractional exponents, which often involve roots as part of their structure.

Fractional Exponents

Fractional exponents (or rational exponents) provide a way to express powers and roots in a unified notation. A fractional exponent like \( a^{m/n} \) consists of:

• Numerator (m): This indicates the power to which the number is raised.
• Denominator (n): Shows the nth root to be taken.

In our exercise, the expression \((-8x^3y^6)^{2/3}\) utilizes fractional exponents.
The exponent \( \frac{2}{3} \) indicates that we take the cube root first (being the root operation of \( 3 \)), following which we square the result. This applies individually to each part of the expression:

• For the term \( (x^3)^{2/3} \): Take the cube root to get \( x \), then square it for \( x^2 \).
• In a similar manner, \( (y^6)^{2/3} \) results in \( y^4 \) after taking the cube root, then elevating it to the power of 2.

Fractional exponents offer a streamlined method to denote operations that would otherwise be more complex if expressed through separate root and power operations.