To add or subtract terms with exponents, both the base and the exponent must be the same. In this case they are so add the coefficients and retain the base and exponent:

-9a⁷ + 8a⁷
(-9 + 8)a⁷
-a⁷

To subtract these radicals together their radicands must be the same:

9\( \sqrt{75} \) - 3\( \sqrt{3} \)
9\( \sqrt{25 \times 3} \) - 3\( \sqrt{3} \)
9\( \sqrt{5^2 \times 3} \) - 3\( \sqrt{3} \)
(9)(5)\( \sqrt{3} \) - 3\( \sqrt{3} \)
45\( \sqrt{3} \) - 3\( \sqrt{3} \)

Now that the radicands are identical, you can subtract them:

To convert a negative exponent to a positive exponent, calculate the positive exponent then take the reciprocal.

A rational number (or fraction) is represented as a ratio between two integers, a and b, and has the form \({a \over b}\) where a is the numerator and b is the denominator. An improper fraction (\({5 \over 3} \)) has a numerator with a greater absolute value than the denominator and can be converted into a mixed number (\(1 {2 \over 3} \)) which has a whole number part and a fractional part.

To simplify this fraction, first find the greatest common factor between them. The factors of 32 are [1, 2, 4, 8, 16, 32] and the factors of 80 are [1, 2, 4, 5, 8, 10, 16, 20, 40, 80]. They share 5 factors [1, 2, 4, 8, 16] making 16 their greatest common factor (GCF).

Next, divide both numerator and denominator by the GCF:

\( \frac{32}{80} \) = \( \frac{\frac{32}{16}}{\frac{80}{16}} \) = \( \frac{2}{5} \)