Problem 30 - Algebra and Trigonometry

Question

Simplify each expression. $$ \left(\frac{3}{x}\right)^{4}\left(\frac{4}{x}\right)^{-2} $$

Step-by-Step Solution

Answer

\( \frac{81}{16x^6} \)

Step 1: Apply the Power of a Quotient Rule

First, apply the power of a quotient rule which states that \( \left( \frac{a}{b} \right)^n = \frac{a^n}{b^n} \). For \( \left( \frac{3}{x} \right)^{4} \), rewrite it as \( \frac{3^4}{x^4} \), which simplifies to \( \frac{81}{x^4} \). For \( \left( \frac{4}{x} \right)^{-2} \), rewrite it as \( \frac{4^{-2}}{x^{-2}} \), which simplifies to \( \frac{1/16}{x^{-2}} \).

Step 2: Simplify the Negative Exponent

Remember that \( x^{-n} = \frac{1}{x^n} \). So, \( x^{-2} \) can be rewritten as \( \frac{1}{x^2} \), allowing \( \frac{1/16}{x^{-2}} \) to become \( \frac{1}{16} \cdot x^2 = \frac{1}{16x^2} \).

Step 3: Multiply the Fractions

Now multiply the expressions \( \frac{81}{x^4} \) and \( \frac{1}{16x^2} \). For multiplication, apply the rule \( \frac{a}{b} \cdot \frac{c}{d} = \frac{a \cdot c}{b \cdot d} \). Thus, \( \frac{81 \cdot 1}{x^4 \cdot 16x^2} = \frac{81}{16x^{6}} \).

Step 4: Simplify the Expression if Possible

Check if there's further simplification possible. Here, all factors are already simplified, and thus the simplified expression is \( \frac{81}{16x^6} \).

Key Concepts

Power of a Quotient Rule

The power of a quotient rule is a helpful exponent rule that simplifies expressions with exponents applied to fractions. It goes by this formula: \( \left( \frac{a}{b} \right)^n = \frac{a^n}{b^n} \). This means you apply the exponent to both the numerator and the denominator.
For example, in the expression \( \left( \frac{3}{x} \right)^4 \), you apply the exponent 4 to both 3 and \( x \), resulting in \( \frac{3^4}{x^4} \). This simplifies further to \( \frac{81}{x^4} \) because \( 3^4 = 81 \). Another example from our problem is \( \left( \frac{4}{x} \right)^{-2} \), which becomes \( \frac{4^{-2}}{x^{-2}} \). This shifts our understanding of the equation and sets up further simplification with negative exponents.

Negative Exponents

Negative exponents can be tricky at first, but they just indicate a reciprocal. The key rule to remember here is that \( x^{-n} = \frac{1}{x^n} \). In essence, a negative exponent flips the base to the opposite side of the fraction line.
Looking at \( x^{-2} \), this translates to \( \frac{1}{x^2} \). By shifting the position from the denominator to the numerator or vice versa, it makes expressions easier to handle and combine. In the exercise, \( \frac{1/16}{x^{-2}} \) becomes \( \frac{1}{16} \cdot x^2 \), which simplifies to \( \frac{1}{16x^2} \). Understanding negative exponents is crucial for tidying up fractions and preparing for multiplication.

Multiplying Fractions

When you're multiplying fractions, the process is straightforward. Just multiply the numerators together and the denominators together. This follows the formula: \( \frac{a}{b} \cdot \frac{c}{d} = \frac{a \cdot c}{b \cdot d} \).
Considering our expressions \( \frac{81}{x^4} \) and \( \frac{1}{16x^2} \), we multiply the numerators 81 and 1, and the denominators \( 16x^2 \) and \( x^4 \), yielding \( \frac{81 \cdot 1}{16 \cdot x^4 \cdot x^2} = \frac{81}{16x^6} \). Keep an eye on both the numbers and variables when multiplying, as combining similar bases through addition of exponents simplifies the result.

Simplifying Expressions

Simplifying expressions means breaking them down into their simplest form. After performing operations like exponentiation, multiplication, or division, check if further reductions are possible.
For the expression \( \frac{81}{16x^6} \), we ensured all numbers and powers of \( x \) are reduced to their simplest form. Once combined, look for common terms or factors that can be canceled out. Here, the numerators and denominators have no further common factors or powers. Therefore, the resultant simple expression reflects clarity and ease for future calculations or comparisons.