Problem 53 - Elementary and Intermediate Algebra

Question

Simplify. \(\left(\frac{c}{d}\right)^{-8}\)

Step-by-Step Solution

Answer

\( \left(\frac{c}{d}\right)^{-8} = \frac{d^8}{c^8} \).

Step 1: Understand Negative Exponents

A negative exponent means that the base is on the wrong side of a fraction line. To simplify, you take the reciprocal of the base and change the sign of the exponent. For example, if you have \( x^{-n} \), it is equivalent to \( \frac{1}{x^n} \).

Step 2: Apply Negative Exponent Rule

Apply the rule for negative exponents to \( \left(\frac{c}{d}\right)^{-8} \). The expression becomes \( \left(\frac{d}{c}\right)^{8} \) because taking the reciprocal of the base cancels out the negative sign of the exponent.

Step 3: Simplifying the Result

Now, express the reciprocal as a simple fraction raised to a positive power: \( \left(\frac{d}{c}\right)^{8} = \frac{d^8}{c^8} \).

Key Concepts

Understanding Fractional Exponents

Fractional exponents may seem complex at first, but they are just another way to express roots in mathematics. When you see a power like \( x^{1/n} \), it represents the \( n \)-th root of \( x \). For instance, \( x^{1/2} \) is the square root of \( x \), and \( x^{1/3} \) is the cube root.
While fractional exponents are primarily about roots, combining base numbers with different fractional exponents requires careful manipulation of powers rather than direct multiplication or division.
In practice:

• If you have \( a^{1/n} \), you can rewrite it as \( \sqrt[n]{a} \).
• When dealing with fractional exponents, rules for integer exponents apply, such as multiplication and division rules, but with the fractional powers.

To simplify expressions involving fractional exponents, remember the basic conversion between radicals and exponents, which will aid in simplifying more complex algebraic expressions.

Grasping the Concept of Reciprocal

The reciprocal of a number or expression is simply flipping it over; in other words, taking "one over" the number. For a number \( x \), its reciprocal is \( \frac{1}{x} \).
When it comes to expressions in algebra, the concept of a reciprocal is useful when dealing with negative exponents. With negative exponents, moving the expression from the numerator to the denominator (or vice versa) makes the exponent positive. For example:

• For a negative exponent, such as \( x^{-1} \), its reciprocal becomes \( \frac{1}{x^1} \).
• More generally, \( x^{-n} = \frac{1}{x^n} \).

When simplifying expressions, especially involving fractions and exponents, understanding reciprocals ensures accuracy in rewriting and transforming expressions correctly.

Mastering Algebraic Simplification

Algebraic simplification is about making expressions as simple as possible while maintaining their original meaning. It involves reducing expressions to their simplest form without altering their value.
This involves a set of rules and tools used in algebra:

• Combining like terms: Terms with the same variables can be combined. For example, \( 2x + 3x = 5x \).
• Using distributive property: This allows you to multiply each term within a bracket by the term outside, e.g., \( a(b + c) = ab + ac \).
• Simplifying exponents: Apply laws of exponents, like \( a^m \times a^n = a^{m+n} \) and \( (a^m)^n = a^{m \cdot n} \).

Algebraic simplification ensures expressions are easier to understand, work with, and solve, making complex mathematics more accessible and manageable.