Question - Simplifying Complex Fraction Expression with Exponents
mathfraction simplificationsimplify complex fractionsexpression with exponentsexponent simplification
Solution:
The expression in the image shows a fraction with a numerator and a denominator, each containing products of numbers raised to powers. Let's simplify it step by step:Numerator: $$36^{-1} \cdot 40^{-1} \cdot 10^2 \cdot 5 \cdot 100$$Denominator: $$2^3 \cdot 14^{-1} \cdot 5 \cdot 25$$First, let's simplify the exponents and the numbers within the expression:$$36^{-1}$$ is the reciprocal of 36, which is $$ \frac{1}{36} $$.$$40^{-1}$$ is the reciprocal of 40, which is $$ \frac{1}{40} $$.$$10^2$$ is $$10 \cdot 10 = 100$$.$$14^{-1}$$ is the reciprocal of 14, which is $$ \frac{1}{14} $$.$$2^3$$ is $$2 \cdot 2 \cdot 2 = 8$$.Now, placing these into the expression, we get:$$\frac{\frac{1}{36} \cdot \frac{1}{40} \cdot 100 \cdot 5 \cdot 100}{8 \cdot \frac{1}{14} \cdot 5 \cdot 25}$$Next, we can simplify the numbers:- $$100 \cdot 5 \cdot 100$$ in the numerator simplifies to $$50,000$$.- In the denominator, $$25$$ is equivalent to $$5^2$$, and since there is a $$5$$ in both the numerator and denominator, they will cancel each other out.- The 5's and the 8 in the denominator can now be combined with the 50,000 in the numerator by division.The expression simplifies:$$\frac{\frac{1}{36} \cdot \frac{1}{40} \cdot 50,000}{8 \cdot \frac{1}{14} \cdot 5^2}$$Dividing 50,000 by 8 gives 6,250:$$\frac{\frac{1}{36} \cdot \frac{1}{40} \cdot 6,250}{\frac{1}{14} \cdot 25}$$Now, multiplying $$\frac{1}{14}$$ by 25 gives $$\frac{25}{14}$$, and we can multiply the fractions in the numerator and denominator:$$\frac{(\frac{1}{36} \cdot \frac{1}{40} \cdot 6,250)}{(\frac{25}{14})}$$Next, we can simplify the 6,250 in the numerator by dividing it by 25 in the denominator, which yields 250:$$\frac{(\frac{1}{36} \cdot \frac{1}{40} \cdot 250)}{\frac{14}{25}}$$Now we have:$$\frac{(\frac{250}{36 \cdot 40})}{(\frac{14}{25})}$$To simplify further, first multiply 36 by 40:$$36 \cdot 40 = 1,440$$So we have:$$\frac{(\frac{250}{1440})}{(\frac{14}{25})}$$Next, to divide by a fraction, you multiply by its reciprocal, so we multiply $$\frac{250}{1440}$$ by the reciprocal of $$\frac{14}{25}$$, which is $$\frac{25}{14}$$, getting:$$\frac{250}{1440} \cdot \frac{25}{14}$$Now, multiply both the numerators and the denominators:$$\frac{250 \cdot 25}{1440 \cdot 14}$$$$\frac{6250}{20160}$$To simplify this fraction to lowest terms, we can divide both the numerator and the denominator by their greatest common divisor (GCD). The GCD of 6250 and 20160 is 10, so when we divide both by 10:$$\frac{625}{2016}$$The fraction $$\frac{625}{2016}$$ is already in its simplest form, as there's no common factor between 625 and 2016 besides 1. Therefore, the expression simplifies to:$$\frac{625}{2016}$$This is the most simplified form of the given expression.
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