Problem 54 - Elementary and Intermediate Algebra

Question

Simplify. \(\left(\frac{a}{x}\right)^{-10}\)

Step-by-Step Solution

Answer

\(\frac{x^{10}}{a^{10}}\)

Step 1: Understand Negative Exponents

A negative exponent indicates the reciprocal of the base. That means for any base \(a\) and exponent \(-n\), \(a^{-n} = \frac{1}{a^n}\). Apply this rule to the given expression: \(\left(\frac{a}{x}\right)^{-10}\). This becomes \(\frac{1}{\left(\frac{a}{x}\right)^{10}}\).

Step 2: Simplify the Fractional Exponent

Now analyze the expression \(\frac{1}{\left(\frac{a}{x}\right)^{10}}\). The exponent \(10\) applies to both the numerator and denominator in the fraction, i.e., \(\frac{a^{10}}{x^{10}}\). Hence, the expression simplifies to its reciprocal: \(\frac{x^{10}}{a^{10}}\).

Step 3: Final Simplification

The expression is now \(\frac{x^{10}}{a^{10}}\), which is in its simplest form. No further simplification is possible since both the numerator and denominator are powers of different bases.

Key Concepts

Negative Exponents

Negative exponents are a crucial part of learning algebraic expressions. They provide a way to express very small numbers in a more manageable form.

• When you see a negative exponent, it simply tells you to take the reciprocal of the base and make the exponent positive.
• For example, with a base of \(b\) and an exponent of \(-n\), the expression becomes \(b^{-n} = \frac{1}{b^n}\).
• In the context of our exercise, \(\left(\frac{a}{x}\right)^{-10}\) indicates taking the reciprocal of \(\left(\frac{a}{x}\right)\) and raising it to the power of 10.

Understanding this basic rule allows you to transform complex expressions into simpler ones by eliminating negative exponents.

Fractional Exponents

Fractional exponents can seem tricky at first, but they are generally extensions of whole number exponents.

• A fractional exponent represents both a power and a root. For instance, \(a^{m/n}\) means \((a^m)^{1/n}\) or, alternatively, \((a^{1/n})^m\).
• When dealing with fractions inside exponents, like \(\left(\frac{a}{x}\right)^{10}\), the exponent applies to both the numerator and the denominator.
• This means the expression \(\frac{1}{\left(\frac{a}{x}\right)^{10}}\) can be rewritten as \(\frac{x^{10}}{a^{10}}\), distributing the power across the fraction.

By working with fractional exponents, you can simplify expressions that would otherwise look complex and cumbersome.

Simplification

Simplifying expressions is a key skill to master in algebra. It involves reducing expressions to their simplest form without changing their value.

• The goal is to make the expressions easier to handle and understand.
• In our example, the expression \(\frac{x^{10}}{a^{10}}\) is simpler than \(\left(\frac{a}{x}\right)^{-10}\) despite representing the same quantity.
• This expression is in its simplest form as it can't be reduced any further.
• Simplifying is not just about working calculations through; it's about recognizing the form and transforming it into something manageable.

By focusing on simplification, you make problem-solving faster and reduce the risk of errors.