Dada la siguiente expresión matemática que se muestra a continuación RADICACIÓN DE NÜMEROS ENTERO
$\left(\frac{2^5 \times 2^2}{2^4}\right)^2=$
Su resultado es:
a. $、 16$
b. 32
c. 64
d. 128
Answer (2)
Simplify the numerator using the property a m × a n = a m + n : 2 5 × 2 2 = 2 7 .
Simplify the fraction using the property a n a m = a m − n : 2 4 2 7 = 2 3 .
Simplify the entire expression using the property ( a m ) n = a m × n : ( 2 3 ) 2 = 2 6 .
Calculate the final result: 2 6 = 64 . The answer is 64 .
Explanation
Understanding the Problem We are given the expression ( 2 4 2 5 × 2 2 ) 2 and we need to find its result.
Simplifying the Numerator First, we simplify the expression inside the parentheses. We know that when multiplying numbers with the same base, we add the exponents: a m × a n = a m + n . Therefore, 2 5 × 2 2 = 2 5 + 2 = 2 7 .
Simplifying the Fraction Now we have ( 2 4 2 7 ) 2 . When dividing numbers with the same base, we subtract the exponents: a n a m = a m − n . Therefore, 2 4 2 7 = 2 7 − 4 = 2 3 .
Simplifying the Entire Expression Now we have ( 2 3 ) 2 . When raising a power to a power, we multiply the exponents: ( a m ) n = a m × n . Therefore, ( 2 3 ) 2 = 2 3 × 2 = 2 6 .
Calculating the Final Result Finally, we calculate 2 6 = 2 × 2 × 2 × 2 × 2 × 2 = 64 .
Conclusion Therefore, the result of the expression is 64.
Examples
Exponents are used to calculate compound interest. For example, if you invest money in a bank account that pays compound interest, the amount of money you have at the end of each year can be calculated using exponents. Also, exponents are used in computer science to calculate the amount of memory needed to store data. For example, the amount of memory needed to store an image is proportional to the square of the image's width and height.
The result of the expression ( 2 4 2 5 × 2 2 ) 2 is calculated to be 64. The steps include simplifying the numerator and denominator, then squaring the result. Hence, the correct option is c .64 .
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