The absolute value is the positive magnitude of a particular number or variable and is indicated by two vertical lines: \(\left|-5\right| = 5\). In the case of a variable absolute value (\(\left|a\right| = 5\)) the value of a can be either positive or negative (a = -5 or a = 5).

To add these fractions, first find the lowest common multiple of their denominators. The first few multiples of 8 are [8, 16, 24, 32, 40, 48, 56, 64, 72, 80] and the first few multiples of 12 are [12, 24, 36, 48, 60, 72, 84, 96]. The first few multiples they share are [24, 48, 72, 96] making 24 the smallest multiple 8 and 12 share.

Next, convert the fractions so each denominator equals the lowest common multiple:

\( \frac{3 x 3}{8 x 3} \) + \( \frac{4 x 2}{12 x 2} \)

\( \frac{9}{24} \) + \( \frac{8}{24} \)

Now, because the fractions share a common denominator, you can add them:

\( \frac{9 + 8}{24} \) = \( \frac{17}{24} \) = \(\frac{17}{24}\)

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

5x⁷ - 4x⁷
(5 - 4)x⁷
x⁷

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

5\( \sqrt{45} \) + 2\( \sqrt{5} \)
5\( \sqrt{9 \times 5} \) + 2\( \sqrt{5} \)
5\( \sqrt{3^2 \times 5} \) + 2\( \sqrt{5} \)
(5)(3)\( \sqrt{5} \) + 2\( \sqrt{5} \)
15\( \sqrt{5} \) + 2\( \sqrt{5} \)

Now that the radicands are identical, you can add them together:

To simplify this fraction, first find the greatest common factor between them. The factors of 20 are [1, 2, 4, 5, 10, 20] and the factors of 48 are [1, 2, 3, 4, 6, 8, 12, 16, 24, 48]. They share 3 factors [1, 2, 4] making 4 their greatest common factor (GCF).

Next, divide both numerator and denominator by the GCF:

\( \frac{20}{48} \) = \( \frac{\frac{20}{4}}{\frac{48}{4}} \) = \( \frac{5}{12} \)