The area of a circle is given by the formula A = πr² where r is the radius of the circle. The radius of a circle is its diameter divided by two so A = π(\( \frac{d}{2} \))². If the diameter of the logo increases by 70% the radius (and, consequently, the total area) increases by \( \frac{70\text{%}}{2} \) = 35%

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 divide fractions, invert the second fraction and then multiply:

\( \frac{1}{8} \) ÷ \( \frac{2}{7} \) = \( \frac{1}{8} \) x \( \frac{7}{2} \)

To multiply fractions, multiply the numerators together and then multiply the denominators together:

\( \frac{1}{8} \) x \( \frac{7}{2} \) = \( \frac{1 x 7}{8 x 2} \) = \( \frac{7}{16} \) = \(\frac{7}{16}\)

To subtract these fractions, first find the lowest common multiple of their denominators. The first few multiples of 9 are [9, 18, 27, 36, 45, 54, 63, 72, 81, 90] and the first few multiples of 15 are [15, 30, 45, 60, 75, 90]. The first few multiples they share are [45, 90] making 45 the smallest multiple 9 and 15 share.

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

\( \frac{4 x 5}{9 x 5} \) - \( \frac{3 x 3}{15 x 3} \)

\( \frac{20}{45} \) - \( \frac{9}{45} \)

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

\( \frac{20 - 9}{45} \) = \( \frac{11}{45} \) = \(\frac{11}{45}\)

To multiply terms with exponents, the base of both exponents must be the same. In this case they are so multiply the coefficients and add the exponents:

-4c⁷ x 8c²
(-4 x 8)c^{(7 + 2)}
-32c⁹