Question

Decide which expressions are equiva lent. Assume all variables are positive. (a) \(\frac{1}{x^{-r}}\) (b) \(\frac{1}{x^{r}}\) (c) \(\left(\frac{1}{x}\right)^{-r}\) (d) \(x^{-r}\) (e) \(\frac{1}{\frac{1}{x^{r}}}\)

Short answer

Answer: The equivalent expressions are (a) and (e), (a) and (c), (b) and (d), and (c) and (e).

Step-by-step solution

Evaluate each expression individually

We need to simplify each expression (a) through (e). (a) \(\frac{1}{x^{-r}}\) Using the rule \(x^{-n} = \frac{1}{x^n}\), this expression simplifies to: \(\frac{1}{\frac{1}{x^r}}\) (b) \(\frac{1}{x^{r}}\) This expression cannot be simplified further. (c) \(\left(\frac{1}{x}\right)^{-r}\) Here, using the rule \(\left(\frac{a}{b}\right)^{-n} = \left(\frac{b}{a}\right)^{n}\), we obtain: \(\left(\frac{x}{1}\right)^{r} = x^{r}\) (d) \(x^{-r}\) Using the rule \(x^{-n} = \frac{1}{x^n}\), this expression simplifies to: \(\frac{1}{x^r}\) (e) \(\frac{1}{\frac{1}{x^{r}}}\) Using the rule \(\frac{a}{\frac{b}{c}} = \frac{ac}{b}\), we obtain: \(\frac{x^r}{1} = x^r\)

Compare simplified expressions

Now, we find the equivalent expressions: (a) \(\frac{1}{\frac{1}{x^r}}\) = (e) \(x^r\) (b) \(\frac{1}{x^{r}}\) = (d) \(\frac{1}{x^r}\) (c) \(x^{r}\) = (e) \(x^r\) In conclusion, the equivalent expressions are (a) and (e), as well as (b) and (d). Expression (c) is also equivalent to (a) and (e).

Key concepts

Negative Exponents

Negative exponents may seem a bit tricky at first, but they're actually quite straightforward once you understand the core idea. A negative exponent means that the base of the power is on the wrong side of the fraction line, so to speak.
When you see an expression like \(x^{-r}\), you can use the rule for negative exponents to rewrite it as \(\frac{1}{x^r}\). This transformation helps in simplifying expressions and finding equivalence in problems involving exponents.
For example, in our exercise, expression \(d) x^{-r}\) simplifies to \(\frac{1}{x^r}\) by this very rule. The same concept applies to expression \(a) \frac{1}{x^{-r}}\), but here the negative exponent moves \(x\) to the numerator when simplified. Keep practicing with negative exponents, and they will become more intuitive.

Expression Equivalence

Expression equivalence means that different looking expressions can actually be equal under certain conditions. In algebra, particularly with exponentiation, two expressions can often be transformed to look different yet be equivalent.
Through simplification, we identify these equivalences. In our exercise:

• Expression \(a) \frac{1}{x^{-r}}\) and expression \(e) x^r\) seem different, but once we utilize the negative exponent rule, they are the same.
• Similarly, expression \(b) \frac{1}{x^{r}}\) is equivalent to \(d) \frac{1}{x^r}\) simply because they are already in the standard form for negative exponent conversion.
• Expression \(c) \left(\frac{1}{x}\right)^{-r}\) also simplifies to \(x^r\), matching expression (e).

This shows that recognizing equivalent expressions is crucial for solving algebraic problems, and practice helps develop this skill.

Simplifying Expressions

Simplifying expressions involves transforming them into the most basic or standard form possible without changing their value. This often involves applying algebraic rules like those for exponents.
Let's take a look at how simplification works in our exercise:

• Expression \(a) \frac{1}{x^{-r}}\) simplifies to \(x^r\), which is much easier to handle than its original form.
• Expression \(b) \frac{1}{x^{r}}\) remains the same because it's already in simplified form, but it's important to recognize that its equivalent, \(d) x^{-r}\), requires simplification.
• Expression \(c) \left(\frac{1}{x}\right)^{-r}\) simplifies directly to \(x^r\) using inversion of the fraction.

These simplifications allow for more straightforward comparison of expressions, critical when determining equivalency or solving more complex problems. Understanding this will streamline the problem-solving process.