Question

Which expressions are in their simplest form? Check all that apply.
One-third + x Superscript 7
x Superscript negative 9 Baseline minus StartFraction 1 Over y EndFraction
StartFraction 1 Over x cubed EndFraction minus StartFraction 1 Over y Superscript 4 EndFraction
x cubed + StartFraction 1 Over y EndFraction minus t Superscript 6
x Superscript negative 5 Baseline minus y Superscript negative 4

Answer 1

To determine whether each of the given expressions is in its simplest form, we can analyze them for any possible simplifications.

1. **One-third + x Superscript 7**:
- This expression cannot be simplified further. **It is in simplest form.**

2. **x Superscript negative 9 Baseline minus StartFraction 1 Over y EndFraction**:
- This expression contains a negative exponent and a fraction. The negative exponent can be rewritten as \( \frac{1}{x^9} \), which gives:
\[
\frac{1}{x^9} - \frac{1}{y}
\]
- As written, it is a mixture of terms, and they cannot be combined. **It is in simplest form.**

3. **StartFraction 1 Over x cubed EndFraction minus StartFraction 1 Over y Superscript 4 EndFraction**:
- This can also be expressed as:
\[
\frac{1}{x^3} - \frac{1}{y^4}
\]
- Since the denominators are different and there are no like terms, **it is in simplest form.**

4. **x cubed + StartFraction 1 Over y EndFraction minus t Superscript 6**:
- This expression contains \( x^3 \), \( \frac{1}{y} \), and \( -t^6 \), which are all distinct terms. There are no like terms to combine. **It is in simplest form.**

5. **x Superscript negative 5 Baseline minus y Superscript negative 4**:
- The negative exponents can be rewritten as:
\[
\frac{1}{x^5} - \frac{1}{y^4}
\]
- As written, it has distinct terms, and they cannot be combined. **It is in simplest form.**

**Summary: All of the provided expressions are in their simplest form.**