Evaluate $\dfrac{10(5+3)}{(2^6-10 \times 6)^2}$ $+$ $5^2$

Ten times five plus three divided by square of two raised to the power six minus ten times six plus five squared is a given arithmetic expression in this problem.

Numerator of first term in exponential form

The first term in the denominator and the second term in the expression are in exponential notation. It indicates that the numerator in the first term should be expressed in exponential form. In order to express numerator in exponential notation, firstly add the numbers five and three in the second factor position of the numerator of first term in the arithmetic expression.

$=\,\,\,$ $\dfrac{10(8)}{(2^6-10 \times 6)^2}$ $+$ $5^2$

$=\,\,\,$ $\dfrac{10 \times 8}{(2^6-10 \times 6)^2}$ $+$ $5^2$

Now, factorize the numbers ten and eight for expressing it in exponential notation.

$=\,\,\,$ $\dfrac{(5 \times 2) \times (2 \times 2 \times 2)}{(2^6-10 \times 6)^2}$ $+$ $5^2$

$=\,\,\,$ $\dfrac{5 \times 2 \times 2 \times 2 \times 2}{(2^6-10 \times 6)^2}$ $+$ $5^2$

Four times the number two is multiplied by itself. So, the product of them can be now written in exponential notation.

$=\,\,\,$ $\dfrac{5 \times 2^4}{(2^6-10 \times 6)^2}$ $+$ $5^2$

Express the product in exponential notation

The second term of the binomial in the denominator of the first term is the product of ten and six. Just like in the first step, factorize the product of them and then express them in exponential form.

$=\,\,\,$ $\dfrac{5 \times 2^4}{(2^6-(5 \times 2) \times (3 \times 2))^2}$ $+$ $5^2$

$=\,\,\,$ $\dfrac{5 \times 2^4}{(2^6-5 \times 2 \times 3 \times 2)^2}$ $+$ $5^2$

$=\,\,\,$ $\dfrac{5 \times 2^4}{(2^6-5 \times 3 \times 2 \times 2)^2}$ $+$ $5^2$

The numbers five and three are not represented in the product. So, multiply them to get their product but two times the number two is repeated in the product. Therefore, write their product in exponential notation.

$=\,\,\,$ $\dfrac{5 \times 2^4}{(2^6-15 \times 2^2)^2}$ $+$ $5^2$

Evaluate the expression by simplification

$=\,\,\,$ $\dfrac{5 \times 2^4}{(2^{2\,+\,4}-15 \times 2^2)^2}$ $+$ $5^2$

$=\,\,\,$ $\dfrac{5 \times 2^4}{(2^2 \times 2^4-15 \times 2^2)^2}$ $+$ $5^2$

$=\,\,\,$ $\dfrac{5 \times 2^4}{\big(2^2 \times (2^4-15)\big)^2}$ $+$ $5^2$

$=\,\,\,$ $\dfrac{5 \times 2^4}{(2^2)^2 \times (2^4-15)^2}$ $+$ $5^2$

$=\,\,\,$ $\dfrac{5 \times 2^4}{2^{2 \times 2} \times (2^4-15)^2}$ $+$ $5^2$

$=\,\,\,$ $\dfrac{5 \times 2^4}{2^4 \times (2^4-15)^2}$ $+$ $5^2$

$=\,\,\,$ $\dfrac{5 \times \cancel{2^4}}{\cancel{2^4} \times (2^4-15)^2}$ $+$ $5^2$

$=\,\,\,$ $\dfrac{5}{(2^4-15)^2}$ $+$ $5^2$

$=\,\,\,$ $\dfrac{5}{(2 \times 2 \times 2 \times 2-15)^2}$ $+$ $5^2$

$=\,\,\,$ $\dfrac{5}{(4 \times 2 \times 2-15)^2}$ $+$ $5^2$

$=\,\,\,$ $\dfrac{5}{(8 \times 2-15)^2}$ $+$ $5^2$

$=\,\,\,$ $\dfrac{5}{(16-15)^2}$ $+$ $5^2$

$=\,\,\,$ $\dfrac{5}{(1)^2}$ $+$ $5^2$

$=\,\,\,$ $\dfrac{5}{1^2}$ $+$ $5^2$

$=\,\,\,$ $\dfrac{5}{1 \times 1}$ $+$ $5^2$

$=\,\,\,$ $\dfrac{5}{1}$ $+$ $5^2$

$=\,\,\,$ $5$ $+$ $5^2$

$=\,\,\,$ $5$ $+$ $5 \times 5$

$=\,\,\,$ $5$ $+$ $25$

$=\,\,\,$ $30$

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