Problem 53 - Algebra and Trigonometry

Question

Simplify the expression and eliminate any negative exponents \((\mathrm{s}) .\) Assume that all letters denote positive numbers. \(\left(2 x^{4} y^{-4 / 5}\right)^{3}\left(8 y^{2}\right)^{2 / 3}\)

Step-by-Step Solution

Answer

The simplified expression is \( 32 x^{12} y^{-16/15} \).

Step 1: Simplify the first expression

First, let's simplify \( \left(2 x^{4} y^{-4/5}\right)^{3} \). Apply the power of a power rule, which states that \( (a^m)^n = a^{mn} \). Thus, \( (2 x^4 y^{-4/5})^3 = 2^3 (x^4)^3 (y^{-4/5})^3 = 8 x^{12} y^{-12/5} \).

Step 2: Simplify the second expression

Now, simplify \( \left(8 y^{2}\right)^{2/3} \). Again, use the power of a power rule: \( (8 y^2)^{2/3} = 8^{2/3} (y^2)^{2/3} \). The cube root of 8 is 2, which gives \( 8^{2/3} = (2^3)^{2/3} = 2^2 = 4 \). Simplify \( (y^2)^{2/3} = y^{4/3} \). Thus, the expression becomes \( 4 y^{4/3} \).

Step 3: Multiply the simplified expressions

Now we combine the simplified expressions: \( 8 x^{12} y^{-12/5} \cdot 4 y^{4/3} \). Combine the coefficients, giving \( 8 \cdot 4 = 32 \). Multiply the like bases by adding exponents: \( x^{12} \) remains as is and for \( y \), add the exponents \( -\frac{12}{5} + \frac{4}{3} \).

Step 4: Add the exponents of y

To sum the exponents \( -\frac{12}{5} + \frac{4}{3} \), first convert them to a common denominator. Convert \(-\frac{12}{5}\) to \(-\frac{36}{15}\) and \(\frac{4}{3}\) to \(\frac{20}{15}\). Adding these, \(-\frac{36}{15} + \frac{20}{15} = -\frac{16}{15} \).

Step 5: Form the final expression

Now, combine all parts to form the final expression: \( 32 x^{12} y^{-16/15} \).

Key Concepts

Negative Exponents

Negative exponents can often seem confusing, but they are quite simple once you understand the basic concept. A negative exponent means that instead of multiplying, you are dividing. Specifically, for any positive number \( a \), \( a^{-n} = \frac{1}{a^n} \). This rule allows us to transform negative exponents into positive ones by rewriting the expression as a fraction.
When simplifying expressions that include negative exponents, we need to convert them to positive exponents if the instructions specify so. For example, \( y^{-4/5} \) becomes \( \frac{1}{y^{4/5}} \). This is a key step in ensuring our final expression is simplified according to the requirement to "eliminate negative exponents."
By recognizing and correctly handling negative exponents, you can simplify complex expressions more effectively.

Power of a Power Rule

The power of a power rule is a fundamental concept when dealing with exponents in mathematics. It states that \( (a^m)^n = a^{mn} \). In simpler terms, when you have an exponent raised to another exponent, you multiply the exponents.
Consider the expression \( (2 x^4 y^{-4/5})^3 \). Applying the power of a power rule, each element inside the parentheses is raised to the third power:

• \( 2^3 = 8 \)
• \( (x^4)^3 = x^{12} \)
• \( (y^{-4/5})^3 = y^{-12/5} \)

Using this rule simplifies expressions significantly by consolidating exponential terms and reducing the complexity of the equation.
Mastering this rule not only helps in simplifying expressions but also speeds up complex calculations dramatically.

Common Denominator

Finding a common denominator is essential when adding or subtracting fractions, especially in algebraic expressions involving exponents. It ensures that the fractions have a common base for easy computation.
In the expression \( 8 x^{12} y^{-12/5} \cdot 4 y^{4/3} \), we encounter two different exponents for \( y \): \(-\frac{12}{5} \) and \( \frac{4}{3} \). To combine these fractions, a common denominator must be found. The denominators here are 5 and 3, and a common denominator for these is 15.
Now, convert the fractions:

• \( -\frac{12}{5} \) becomes \( -\frac{36}{15} \)
• \( \frac{4}{3} \) becomes \( \frac{20}{15} \)

Adding these fractions with a common denominator gives \( -\frac{16}{15} \), which simplifies the exponent for \( y \).
Developing the skill to find and use a common denominator will greatly aid in solving algebraic expressions involving fractions and exponents.