Problem 47 - Algebra 2 and Trigonometry

Question

In \(35-63,\) write each expression with only positive exponents and express the answer in simplest form. The variables are not equal to zero. $$ \frac{1}{x^{-3}} $$

Step-by-Step Solution

Answer

The expression simplifies to \( x^3 \).

Step 1: Recognizing the Negative Exponent

The given expression is \( \frac{1}{x^{-3}} \). Here, \( x^{-3} \) in the denominator indicates that the exponent of \( x \) needs to be positive. We know from the laws of exponents that a negative exponent in the denominator can be made positive by transferring it to the numerator.

Step 2: Applying the Negative Exponent Rule

Apply the rule for negative exponents which states \( a^{-n} = \frac{1}{a^n} \) or \( \frac{1}{a^{-n}} = a^n \). Thus, \( \frac{1}{x^{-3}} = x^{3} \). By transferring \( x^{-3} \) to the numerator, we change its power from negative to positive.

Step 3: Simplifying the Expression

After applying the rule, the expression is simplified to \( x^3 \). There are no further steps required as the expression already has all positive exponents and is in simplest form.

Key Concepts

Understanding the Laws of Exponents

The laws of exponents give us a handy way to manipulate powers, making complex expressions easier to handle. They are like guidelines that help us work with powers effectively and include rules such as:

• Multiplying powers with the same base: When you multiply two exponents with the same base, you add the powers, i.e., \( a^m \times a^n = a^{m+n} \).
• Dividing powers with the same base: When you divide two exponents with the same base, you subtract the powers, i.e., \( \frac{a^m}{a^n} = a^{m-n} \).
• Raising a power to another power: When you raise an exponent to another exponent, you multiply the powers, i.e., \( (a^m)^n = a^{m \times n} \).
• Any number to the zero power is one: \( a^0 = 1 \), if \( a eq 0 \).

These rules are straightforward ways to simplify your calculations. Familiarize yourself with them, and you'll find algebraic expressions much easier to tackle. When it comes to negative exponents, understanding these fundamentals is essential.

Simplifying Expressions with Exponents

Simplifying expressions is about rewriting them in a more manageable or understood way. In the given exercise, we simplified \( \frac{1}{x^{-3}} \) by turning it into \( x^3 \).This demonstrates a key goal of simplification: expressing the equation in its simplest, positive exponent form.
Simplifying helps in solving equations more efficiently, saving time and reducing potential for errors. Here’s how it can be done:

• Convert negative exponents: Change negative exponents by applying the relevant laws of exponents, as we did in this exercise. For example, \( a^{-1} \) becomes \( \frac{1}{a} \).
• Combine like terms: Use the laws to combine similar terms, reducing clutter in the expression.
• Look for opportunities to apply laws you've learned, like factoring or expanding using exponent rules.

With practice, simplifying becomes habitual, offering clarity and precision in working with algebraic expressions.

Negative Exponents Explained

Negative exponents might look tricky but they're simply a part of the number's power that indicates reciprocal action.
Let's break it down: a negative exponent suggests that the base is repeated as a reciprocal. For instance, \( a^{-n} \) translates to \( \frac{1}{a^n} \).
It's important to understand that:

• Moving terms with negative exponents: From numerator to denominator turns the exponent positive, and vice versa.
• Interpreting negative exponents using the given exercise: The expression \( \frac{1}{x^{-3}} \) is equivalent to \( x^3 \), demonstrating how moving the base \( x^{-3} \) from denominator to numerator reverses the sign.
• Negative exponents don't make the number negative: They only indicate positions in fractions or decimals.

Understanding and correctly applying negative exponents will simplify your math work significantly, helping turn daunting expressions into friendly calculations.