Question

Write the expression as an equivalent expression in the form \(x^{n}\) and give the value for \(n\). $$ 1 /\left(1 / x^{-2}\right) $$

Short answer

Answer: The simplified expression is \(x^2\), and the value of \(n\) is 2.

Step-by-step solution

Rewrite the expression using exponent properties

In order to rewrite the expression, let's start by looking at the inner denominator, which is \(x^{-2}\). Using the properties of exponents, we know that \(x^{-2} = \frac{1}{x^2}\). So, we can rewrite the expression as: $$\frac{1}{\frac{1}{x^2}}$$

Simplify the complex fraction

To simplify the complex fraction, we can use the property: $$\frac{a}{\frac{b}{c}} = a \cdot \frac{c}{b}$$ Applying this property to our expression, we get: $$1 \cdot \frac{x^2}{1}$$

Final simplification

Now, since multiplying by 1 doesn't change the value of a number, the final expression is simply: $$x^2$$

Identify the value of \(n\)

Since the rewritten expression is in the form \(x^{n}\), with \(n = 2\), the value of \(n\) is: $$n = 2$$

Key concepts

Properties of Exponents

Understanding the properties of exponents is fundamental to working with powers and simplifying expressions. When you see an expression like \( x^{-2} \), it means that you are dealing with a reciprocal. The rule

• \( x^{-a} = \frac{1}{x^a} \)

is a powerful tool that allows us to transform negative exponents into fractions with positive exponents. This is what happens in the first step of our problem, where \( x^{-2} \) becomes \( \frac{1}{x^2} \).
This transformation helps us simplify expressions by getting rid of negative exponents, making further operations more straightforward. Exponent rules also tell us that multiplying powers with the same base is straightforward: you simply add the exponents. However, in our exercise, we focused on converting and simplifying the expression by applying this fundamental reciprocal property of exponents.

Simplifying Fractions

Simplifying complex fractions is a crucial skill in algebra. A complex fraction is essentially a fraction where the numerator, the denominator, or both, are also fractions. To simplify, you can use the rule:

• \( \frac{a}{\frac{b}{c}} = a \cdot \frac{c}{b} \)

This means you multiply the top by the reciprocal of the bottom. In our example, \( \frac{1}{\frac{1}{x^2}} \) was simplified by multiplying 1 by the reciprocal of \( \frac{1}{x^2} \), which is \( x^2 \).
This operation allows you to "flip" the bottom part of the fraction and convert the division into multiplication, which often makes the expression much simpler. Such simplifications are key to dealing with algebraic expressions involving fractions by converting them into simpler, more manageable forms.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and operations, such as addition, subtraction, multiplication, and division. In our case, the expression \( \frac{1}{\left( 1 / x^{-2} \right)} \) is complex at first glance, but by applying algebraic rules, it becomes much simpler.
Algebraic manipulation, such as recognizing and applying the properties of exponents and fractions, allows us to simplify or rewrite expressions into more manageable forms like \( x^2 \).
Another important aspect of handling algebraic expressions is identifying equivalent expressions. This means different-looking expressions can actually be equal in value. Our final simplified form \( x^2 \) is equivalent to the original one, demonstrating how algebraic techniques transform and simplify complex-looking problems.