Usando as propriedades simplifique: [tex]\frac{4^{10} \cdot 4^{30}}{4^5 \cdot 4^{15}}=\frac{4}{4}=[/tex]
Answer (2)
To simplify the expression 4 5 ⋅ 4 15 4 10 ⋅ 4 30 , we find that it equals 4 20 after applying the properties of exponents. The expression simplifies to 4 20 which is not equal to 1 . Therefore, the simplified expression results in 4 20 .
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Simplify the numerator using the exponent property a m \t ⋅ a n = a m + n : 4 10 \t ⋅ 4 30 = 4 40 .
Simplify the denominator using the exponent property a m \t ⋅ a n = a m + n : 4 5 \t ⋅ 4 15 = 4 20 .
Divide the numerator by the denominator using the exponent property a n a m = a m − n : 4 20 4 40 = 4 20 .
The simplified expression is 4 20 .
Explanation
Understanding the problem We are asked to simplify the expression 4 5 \t ⋅ 4 15 4 10 \t ⋅ 4 30 = 4 4 = . This involves using the properties of exponents to simplify the left-hand side and then comparing it to the right-hand side.
Simplifying the numerator First, let's simplify the numerator of the left-hand side. Using the property a m ⋅ a n = a m + n , we have 4 10 ⋅ 4 30 = 4 10 + 30 = 4 40
Simplifying the denominator Next, let's simplify the denominator of the left-hand side. Using the same property, we have 4 5 ⋅ 4 15 = 4 5 + 15 = 4 20
Dividing numerator by denominator Now, let's divide the simplified numerator by the simplified denominator. Using the property a n a m = a m − n , we have 4 20 4 40 = 4 40 − 20 = 4 20
Comparing both sides Now, let's simplify the right-hand side of the equation: 4 4 = 1 .
So, the original equation is 4 5 ⋅ 4 15 4 10 ⋅ 4 30 = 4 4 . After simplification, we have 4 20 = 1 . However, 4 20 is a very large number (1099511627776) and is not equal to 1. Therefore, the two sides of the original equation are not equal.
Final Answer The left side simplifies to 4 20 . The right side simplifies to 1. Therefore, the simplified expression for the left side is 4 20 .
Examples
Understanding exponents is crucial in many fields, such as computer science where data sizes are measured in powers of 2 (e.g., kilobytes, megabytes, gigabytes). Simplifying expressions with exponents helps in calculating storage requirements or processing speeds. For instance, if a file is 2 10 bytes and you want to compress it to 4 1 its size, you would calculate 4 2 10 = 2 2 2 10 = 2 10 − 2 = 2 8 bytes.
