የ9ኛ ክፍል ሒሳብ ምዕራፍ 2፦ የቁጥር ሥርዓት

7 is not a perfect square, so √7 has a decimal that never ends and never repeats. √16 equals 4, which is rational. A repeating decimal is always rational, so 0.3333... is 1/3. And 22/7 is a fraction, so it is rational, even though it is often used as an approximation for π.

Split 50 into a perfect square times something else: 50 = 25 x 2. Then √50 = √25 x √2 = 5√2. Option A comes from forgetting to take the square root of 25.

Let x = 0.363636... Two digits repeat, so 100x = 36.363636... Subtracting gives 99x = 36, so x = 36/99. Divide top and bottom by 9 to get 4/11. Option A is right before simplifying, and the question asks for lowest terms.

Multiply top and bottom by √3: (6 x √3) / (√3 x √3) = 6√3 / 3 = 2√3. Option B forgets to divide by the 3 that appears in the denominator.

36 = 2² x 3² and 60 = 2² x 3 x 5. Take the shared primes at their lowest powers: 2² x 3 = 12. Option D is the lowest common multiple, which takes the highest power of every prime instead.

Move the point four places to the right to reach 5.6, so the power is negative four. Options C and D have the right value but the wrong form, because scientific notation needs the first number to be at least 1 and below 10.

Like surds add and subtract the way like terms do in algebra: 3 + 5 - 1 = 7, so the answer is 7√2. Option B forgets to subtract the last term, which counts as 1√2.

The real numbers are defined as the rationals together with the irrationals. The integers, naturals and whole numbers are all subsets of the rationals, so none of them can contain √2.

Multiply top and bottom by the conjugate 4 - √3. The denominator becomes 4² - (√3)² = 16 - 3 = 13, so the answer is (4 - √3) / 13. Option B uses the same sign instead of the conjugate, which leaves a surd in the denominator.

Leading zeros are not significant, so the figures start at 4. The first three are 4, 5 and 6, and the next digit is 7, which rounds the 6 up to 7. That gives 0.00457. Option C keeps the 6 without rounding.