Question

Rewrite the expression with positive exponents and simplify.\(\left(\frac{x}{10}\right)^{-1}\)

Short answer

The expression \(\left(\frac{x}{10}\right)^{-1}\) rewritten with positive exponents and simplified is \(\frac{10}{x}\).

Step-by-step solution

Apply the Negative Exponent Rule

Apply the rule of negative exponents. We know that \(a^{-n} = \frac{1}{a^n}\). So, \( \left( \frac{x}{10} \right)^{-1} = \frac{1}{\left( \frac{x}{10} \right)} \)

Simplify the Fraction

Simplify the fraction within the fraction to rewrite it in a more convenient form. Remember that \(\frac{1}{\frac{a}{b}}\) is equal to \(\frac{b}{a}\). So, \(\frac{1}{\left( \frac{x}{10} \right)} = \frac{10}{x}\)

Key concepts

Negative Exponents

In algebra, negative exponents can seem complicated, but once you understand them, they are a powerful tool. The rule for negative exponents is straightforward: for any non-zero number \( a \) and any integer \( n \), \( a^{-n} = \frac{1}{a^n} \). What does that mean? Essentially, a negative exponent indicates that you take the reciprocal of the base raised to the positive exponent.
For example, if you have \( a^{-1} \), it's the same as \( \frac{1}{a^1} \). This concept helps when you need to simplify expressions or solve equations with variables in the denominator.

Consider dropping the negativity of an exponent by using this reciprocal process; it can make your equations much simpler to handle.

Simplifying Fractions

Simplifying fractions involves making them as simple as possible, by reducing them to their simplest form or converting complex fractions into more manageable pieces. When you encounter a fraction of a fraction, like \( \frac{1}{\left( \frac{x}{10} \right)} \), you can simplify it using a familiar mathematical concept: the reciprocal.
Remember, \( \frac{1}{\left( \frac{a}{b} \right)} = \frac{b}{a} \). So, by flipping the fraction within the fraction, we can directly simplify \( \frac{1}{\left( \frac{x}{10} \right)} \) to \( \frac{10}{x} \).

When simplifying, always aim to reduce fractions to their most elementary form. This makes solving for variables or computations much easier. Keeping fractions simple not only aids in clarity but also helps in making fewer mistakes during calculations.

Exponent Rules

Exponent rules serve as essential guidelines for working with powers and can make operations involving exponents much more manageable. There are several key rules to remember:

• Product of Powers Rule: \( a^m \times a^n = a^{m+n} \)
• Quotient of Powers Rule: \( \frac{a^m}{a^n} = a^{m-n} \) when \( a eq 0 \)
• Power of a Power Rule: \( (a^m)^n = a^{m \times n} \)
• Zero Exponent Rule: \( a^0 = 1 \) for any \( a eq 0 \)
• Negative Exponent Rule: As we've discussed, \( a^{-n} = \frac{1}{a^n} \)

Understanding these rules is critical for correctly manipulating algebraic expressions and for simplifying Algebra problems efficiently. If you take the time to practice and commit these rules to memory, you'll find evaluating expressions with exponents becomes quicker and more intuitive.