Write each of the following square roots in the form a√b, where a and b are integers, b > 0, and b has the least value possible:
a) √180
b) √363
c) √252
a) √180 = √(36 x 5) = 6√5
b) √363 is already in its simplest form since it cannot be simplified further.
c) √252 = √(36 x 7) = 6√7
a) To simplify √180 in the form a√b, we need to find the largest perfect square that divides 180. The largest perfect square that divides 180 is 36 (6^2).
So, √180 can be written as √36 × √5.
Simplifying further, we have 6√5.
Therefore, √180 in the form a√b, where a and b are integers, b > 0, and b has the least value possible, is 6√5.
b) To simplify √363, we need to find the largest perfect square that divides 363. The largest perfect square that divides 363 is 121 (11^2).
So, √363 can be written as √121 × √3.
Simplifying further, we have 11√3.
Therefore, √363 in the form a√b, where a and b are integers, b > 0, and b has the least value possible, is 11√3.
c) To simplify √252, we need to find the largest perfect square that divides 252. The largest perfect square that divides 252 is 36 (6^2).
So, √252 can be written as √36 × √7.
Simplifying further, we have 6√7.
Therefore, √252 in the form a√b, where a and b are integers, b > 0, and b has the least value possible, is 6√7.
To write each of the given square roots in the form a√b, where a and b are integers, b > 0, and b has the least value possible, we need to find the perfect square factors of the numbers inside the square roots.
a) √180:
To find the perfect square factors of 180, we can factorize it into its prime factors: 180 = 2 * 2 * 3 * 3 * 5. Among these factors, the pairs of identical prime factors are 2 * 2 and 3 * 3. We can rewrite the square root of 180 as the square root of the product of the perfect square factors: √(2 * 2 * 3 * 3 * 5) = 2 * 3 √5. Therefore, √180 is written in the form a√b as 6√5, where a = 6 and b = 5.
b) √363:
To find the perfect square factors of 363, we can factorize it: 363 = 3 * 11 * 11. Among these factors, the pair of identical prime factors is 11 * 11. We can rewrite the square root of 363 as the square root of the product of the perfect square factors: √(3 * 11 * 11) = 11 √(3). Therefore, √363 is written in the form a√b as 11√3, where a = 11 and b = 3.
c) √252:
To find the perfect square factors of 252, we can factorize it: 252 = 2 * 2 * 3 * 3 * 7. Among these factors, the pairs of identical prime factors are 2 * 2 and 3 * 3. We can rewrite the square root of 252 as the square root of the product of the perfect square factors: √(2 * 2 * 3 * 3 * 7) = 2 * 3 √7. Therefore, √252 is written in the form a√b as 6√7, where a = 6 and b = 7.
Hence, the square roots are written in the form a√b as follows:
a) √180 = 6√5
b) √363 = 11√3
c) √252 = 6√7
look for the highest perfect square as a factor
a) √180 = √36 x 5 = 6√5
do the others the same way.