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 radicals together their radicands must be the same:

3\( \sqrt{63} \) + 5\( \sqrt{7} \)
3\( \sqrt{9 \times 7} \) + 5\( \sqrt{7} \)
3\( \sqrt{3^2 \times 7} \) + 5\( \sqrt{7} \)
(3)(3)\( \sqrt{7} \) + 5\( \sqrt{7} \)
9\( \sqrt{7} \) + 5\( \sqrt{7} \)

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

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:

7z⁴ + 4z⁴
(7 + 4)z⁴
11z⁴

The equation for this sequence is:

a_n = a_n-1 + 3(n - 1)

where n is the term's order in the sequence, a_n is the value of the term, and a_n-1 is the value of the term before a_n. This makes the next number:

a₆ = a₅ + 3(6 - 1)
a₆ = 31 + 3(5)
a₆ = 46

The distributive property for division helps in solving expressions like \({b + c \over a}\). It specifies that the result of dividing a fraction with multiple terms in the numerator and one term in the denominator can be obtained by dividing each term individually and then totaling the results: \({b + c \over a} = {b \over a} + {c \over a}\). For example, \({a^3 + 6a^2 \over a^2} = {a^3 \over a^2} + {6a^2 \over a^2} = a + 6\).