Verified Answer :
- An expression is in its simplest form when it avoids negative exponents.
- We evaluate each expression: 1, 2, and 5 use positive exponents or fractional notation, while 3 and 4 use negative exponents.
- The expressions in simplest form are:
- \(\frac{1}{x^3} - \frac{1}{y^4}\)
- \(x^3 + \frac{1}{y} - t^6\)
- \(\frac{1}{3} + x^7\)
- Final answer: \(\boxed{1, 2, 5}\)
Explanation
- Understanding Algebraic Simplification
To determine which of the given expressions are in their simplest form, we apply standard algebraic conventions. In algebraic simplification, an expression is generally considered to be in its simplest form if it does not contain negative exponents. Instead, terms with negative exponents are typically rewritten using fractions (i.e., \(x^{-n} = \frac{1}{x^n}\)). We will examine each expression individually based on this criterion.
- Evaluating Each Expression
We evaluate each expression:
- \(\frac{1}{x^3} - \frac{1}{y^4}\): This expression contains only positive exponents in the denominators and no negative exponents. Therefore, it is in its simplest form.
- \(x^3 + \frac{1}{y} - t^6\): This expression is written using a fraction for the term with a negative exponent (instead of \(y^{-1}\)), which is considered the standard simplest form.
- \(x^{-9} - \frac{1}{y}\): This contains the term \(x^{-9}\), which uses a negative exponent. It is not in its simplest form.
- \(x^{-5} - y^{-4}\): This contains the terms \(x^{-5}\) and \(y^{-4}\), both of which use negative exponents. It is not in its simplest form.
- \(\frac{1}{3} + x^7\): This expression contains only positive exponents and constants. It is in its simplest form.
- Conclusion
Based on our evaluation, expressions 1, 2, and 5 do not contain negative exponents and thus represent the expressions in their simplest form. These are:
- \(\frac{1}{x^3} - \frac{1}{y^4}\)
- \(x^3 + \frac{1}{y} - t^6\)
- \(\frac{1}{3} + x^7\)
Examples
Understanding the simplest form of an algebraic expression is useful in many real-world applications, such as physics and engineering, where equations must be manipulated to solve for unknown variables. For example, if you are modeling the velocity of an object, you might start with an expression like \(v = d \cdot t^{-1}\). Rewriting this as \(v = \frac{d}{t}\) makes the relationship much clearer and easier to use for calculations. Mastering these conventions ensures that you can communicate and manipulate mathematical models effectively.