Find the total number of prime factors in the expression \(4^{11} \times 7^5 \times 11^2\).
Answer: A
First, express all bases as prime numbers.
\(4^{11} = (2^2)^{11} = 2^{22}\).
The expression is \(2^{22} \times 7^5 \times 11^2\).
The total number of prime factors is the sum of the exponents of the prime bases.
Total prime factors = 22 + 5 + 2 = 29.
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What is the value of \(587 \times 999\)?
Answer: A
We can write 999 as (1000 - 1).
So, \(587 \times 999 = 587 \times (1000 - 1)\).
= \(587 \times 1000 - 587 \times 1\).
= 587000 - 587 = 586413.
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Find the value of \((243)^{0.16} \times (243)^{0.04}\).
Answer: B
When multiplying powers with the same base, we add the exponents.
The expression becomes \((243)^{0.16 + 0.04} = (243)^{0.20}\).
\(0.20\) as a fraction is \(20/100 = 1/5\).
So we need to find \((243)^{1/5}\), which is the fifth root of 243.
We know that \(3^5 = 3 \times 3 \times 3 \times 3 \times 3 = 243\).
Therefore, the value is 3.
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What is 25% of 50% of 200?
Answer: B
First, calculate 50% of 200.
50% of 200 = \(0.5 \times 200 = 100\).
Now, calculate 25% of 100.
25% of 100 = 25.
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The sum of the digits of a 2-digit number is 9. When 27 is added to the number, the digits get reversed. The number is:
Answer: B
Let the ten's digit be x and the unit's digit be y. Number = 10x + y.
Given: x + y = 9.
Given: (10x + y) + 27 = 10y + x (reversed number).
\(9x - 9y = -27\), which simplifies to \(x - y = -3\) or \(y - x = 3\).
We have two equations: x + y = 9 and y - x = 3.
Adding them: 2y = 12, so y = 6.
Substituting y=6 into x+y=9, we get x=3.
The number is 36.
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What is the smallest number that must be added to 2203 to make it a perfect square?
Answer: C
We need to find the nearest perfect square greater than 2203.
We know \(40^2 = 1600\) and \(50^2 = 2500\). The root is between 40 and 50.
Let's try 47: \(47^2 = 2209\).
The next perfect square after 2203 is 2209.
The number to be added is \(2209 - 2203 = 6\).
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Find the value of \(0.5 \times 0.05 \times 500\).
Answer: B
We can multiply sequentially.
\(0.5 \times 0.05 = 0.025\).
Now, \(0.025 \times 500 = 25 \times 0.5 = 12.5\).
Alternatively, write as fractions: \((\frac{1}{2}) \times (\frac{5}{100}) \times 500 = (\frac{1}{2}) \times (\frac{1}{20}) \times 500 = \frac{500}{40} = \frac{50}{4} = 12.5\).
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If a number is decreased by 4 and divided by 6, the result is 8. What would be the result if 2 is subtracted from the number and then it is divided by 5?
Answer: B
Let the number be x.
According to the first condition: \(\frac{x-4}{6} = 8\).
\(x-4 = 48\), so \(x = 52\).
Now, according to the second condition, subtract 2 from the number: \(52 - 2 = 50\).
Then divide by 5: \(50 \div 5 = 10\).
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Simplify: \(2 \frac{1}{2} + 3 \frac{1}{4} - 1 \frac{1}{3}\)
Answer: A
First, convert the mixed fractions to improper fractions.
\(2 \frac{1}{2} = \frac{5}{2}\)
\(3 \frac{1}{4} = \frac{13}{4}\)
\(1 \frac{1}{3} = \frac{4}{3}\)
The expression is \(\frac{5}{2} + \frac{13}{4} - \frac{4}{3}\).
The LCM of the denominators (2, 4, 3) is 12.
\(\frac{5 \times 6}{12} + \frac{13 \times 3}{12} - \frac{4 \times 4}{12} = \frac{30 + 39 - 16}{12} = \frac{53}{12}\).
Converting back to a mixed fraction: \(53 \div 12\) gives a quotient of 4 and a remainder of 5. So, \(4 \frac{5}{12}\).
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The difference between a number and its three-fifths is 50. What is the number?
Answer: C
Let the number be x.
The equation is \(x - \frac{3}{5}x = 50\).
\(\frac{5x-3x}{5} = 50\).
\(\frac{2}{5}x = 50\).
\(x = 50 \times \frac{5}{2} = 25 \times 5 = 125\).
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