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Simplify: $\newline$ $(i) \frac{(2^{5})^{2}\times 7^{3}}{8^{3}\times 7}$

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Gaussian integral

Full solution

Q. Simplify:

\newline

(i) \frac{(2^{5})^{2}\times 7^{3}}{8^{3}\times 7}

Simplify Terms Separately: Simplify the numerator and the denominator separately. $\newline$ The numerator is $(2^5)^2 \times 7^3$ . $\newline$ The denominator is $8^3 \times 7$ .
Apply Power Rule: Apply the power of a power rule to the term $(2^5)^2$ . According to the power of a power rule, $(a^b)^c = a^{(b*c)}$ . So, $(2^5)^2 = 2^{(5*2)} = 2^{10}$ .
Recognize Power of $2$ : Recognize that $8$ is a power of $2$ , specifically $8 = 2^3$ . So, $8^3$ can be written as $(2^3)^3$ . Applying the power of a power rule again, we get $(2^3)^3 = 2^{(3*3)} = 2^9$ .
Rewrite with Simplified Terms: Rewrite the original expression with the simplified terms. $\newline$ The expression now becomes $(2^{10} \times 7^{3}) / (2^{9} \times 7)$ .
Cancel Common Terms: Cancel out common terms in the numerator and the denominator. $\newline$ We can cancel out $2^9$ from the numerator and the denominator, leaving us with $2^{10-9}$ in the numerator. $\newline$ We can also cancel out $7$ from the numerator and the denominator, leaving us with $7^{3-1}$ in the numerator. $\newline$ The expression now becomes $2^1 \times 7^2$ .
Calculate Remaining Expression: Calculate the remaining expression. $\newline$ $2^1 = 2$ and $7^2 = 49$ . $\newline$ So, the simplified expression is $2 \times 49$ .
Multiply for Final Answer: Multiply the remaining terms to get the final answer. $2 \times 49 = 98$ .

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