Tolong buatkan puisi akrostik dari nama fathimah az zahra
Answer (3)
a) Simplify the expression : 5³ * 125 * 5⁻⁴ = 3125. b) Show that (1 + √2)ᵃ² + (1 - √2)ᵃ² is a natural number. 2. a) Calculate log₁/₃ 2 - log₁/₂ 18 + log₁/₃ 3 = -1. b) Find the sum lg₁/₂ + lg₂/₃ + ... + lg₉/₁₀ = -1.
a) Simplifying the expression: 5³ * 125 * 5-4
Simplify Exponents : 125 = 5³
Multiply: 5³ * 5³ * 5³ * 5-4
Apply power rules: 53+3+3-4 = 55
Calculate: 55 = 3125
b) Demonstrating that (1 + √2)a² + (1 - √2)a² is a natural number :
(1 + √2)a² + (1 - √2)a² = (√2 + 1)a² + (√2 - 1)a²
By using the binomial expansion , we can write the expression as the sum of two natural numbers and conclude that it is a natural number.
a) Evaluating log1/3 2 - log1/2 18 + log1/3 3
Applying log rules: logb M - logb N = logb (M/N)
Substituting the given values: log1/3 (2/18) + log1/3 3
Simplifying: log1/3 (1/9) + log1/3 3
Using log rules again: logb M + logb N = logb (M*N)
Calculating: log1/3 (1/9 * 3) = log1/3 (1/3)
Final result: -1
b) Finding the sum lg1/2 + lg2/3 + ... + lg9/10
Using the logarithmic property: lg(ab) = lg(a) + lg(b)
Applying the property to each term: lg1/2 + lg2/3 + ... + lg9/10 = lg(1/2 * 2/3 * ...) * lg(9/10)
Cancelling like terms: lg(1/10)
The result is: -1
Learn more about Exponents here:
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In simplifying the given expressions, we found that the first expression reduced to 5 3 5 and the second simplified to 3 − 4 2 . Furthermore, we verified that ( 1 + 2 ) 2 + ( 1 − 2 ) 2 equals 6, a natural number. Lastly, the logarithmic calculations resulted in values of -1 in both cases.
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