Convert \( 321_4 \) to base 10.
A. 49
B. 57
C. 55
D. 59
Answer: B) 57
Explanation:
\( 321_4 = 3 \times 4^2 + 2 \times 4^1 + 1 \times 4^0 \)
\( = 3 \times 16 + 2 \times 4 + 1 \times 1 \)
\( = 48 + 8 + 1 = 57 \).
Express \( 0.375 \) as a fraction in its simplest form.
A. \( \frac{3}{8} \)
B. \( \frac{3}{5} \)
C. \( \frac{5}{8} \)
D. \( \frac{3}{10} \)
Answer: A) \( \frac{3}{8} \)
Explanation:
\( 0.375 = \frac{375}{1000} \).
Simplify by dividing numerator and denominator by 125:
\( \frac{375 \div 125}{1000 \div 125} = \frac{3}{8} \).
If \( 27^x = 9 \), find \( x \).
A. \( \frac{2}{3} \)
B. \( \frac{3}{2} \)
C. 2
D. \( \frac{1}{2} \)
Answer: A) \( \frac{2}{3} \)
Explanation:
Rewrite \( 27 \) and \( 9 \) as powers of 3:
\( 27^x = 9 \) ⇒ \( (3^3)^x = 3^2 \) ⇒ \( 3^{3x} = 3^2 \).
Equate the exponents: \( 3x = 2 \) ⇒ \( x = \frac{2}{3} \).
If \( A = \{2, 3, 4, 5\} \) and \( B = \{4, 5, 6, 7\} \), find \( A \cap B \).
A. \( \{6, 7\} \)
B. \( \{2, 3, 6, 7\} \)
C. \( \{4, 5\} \)
D. \( \{2, 3\} \)
Answer: C) \( \{4, 5\} \)
Explanation:
The intersection \( A \cap B \) contains elements common to both sets: \( \{4, 5\} \).
Factorize \( x^2 - 9x + 20 \).
A. \( (x - 5)(x - 4) \)
B. \( (x + 5)(x + 4) \)
C. \( (x - 5)(x + 4) \)
D. \( (x + 5)(x - 4) \)
Answer: A) \( (x - 5)(x - 4) \)
Explanation:
\( x^2 - 9x + 20 = (x - 5)(x - 4) \), since \( -5 \) and \( -4 \) multiply to 20 and add to \( -9 \).