To subtract these fractions, first find the lowest common multiple of their denominators. The first few multiples of 6 are [6, 12, 18, 24, 30, 36, 42, 48, 54, 60] and the first few multiples of 12 are [12, 24, 36, 48, 60, 72, 84, 96]. The first few multiples they share are [12, 24, 36, 48, 60] making 12 the smallest multiple 6 and 12 share.

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

\( \frac{7 x 2}{6 x 2} \) - \( \frac{9 x 1}{12 x 1} \)

\( \frac{14}{12} \) - \( \frac{9}{12} \)

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

\( \frac{14 - 9}{12} \) = \( \frac{5}{12} \) = \(\frac{5}{12}\)

To simplify this fraction, first find the greatest common factor between them. The factors of 24 are [1, 2, 3, 4, 6, 8, 12, 24] and the factors of 72 are [1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72]. They share 8 factors [1, 2, 3, 4, 6, 8, 12, 24] making 24 their greatest common factor (GCF).

Next, divide both numerator and denominator by the GCF:

\( \frac{24}{72} \) = \( \frac{\frac{24}{24}}{\frac{72}{24}} \) = \( \frac{1}{3} \)

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%

To add these fractions, first find the lowest common multiple of their denominators. The first few multiples of 4 are [4, 8, 12, 16, 20, 24, 28, 32, 36, 40] and the first few multiples of 10 are [10, 20, 30, 40, 50, 60, 70, 80, 90]. The first few multiples they share are [20, 40, 60, 80] making 20 the smallest multiple 4 and 10 share.

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

\( \frac{9 x 5}{4 x 5} \) + \( \frac{9 x 2}{10 x 2} \)

\( \frac{45}{20} \) + \( \frac{18}{20} \)

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

\( \frac{45 + 18}{20} \) = \( \frac{63}{20} \) = 3\(\frac{3}{20}\)

A number with an exponent (b^e) consists of a base (b) raised to a power (e). The exponent indicates the number of times that the base is multiplied by itself. A base with an exponent of 1 equals the base (b¹ = b) and a base with an exponent of 0 equals 1 ( (b⁰ = 1).