Numbers and Operations - ISEE Upper Level (grades 9-12) Quantitative Reasoning

The sum of the first ten perfect square integers:

The sum of the first five perfect cube integers:

(A) is greater.

The perfect squares between 50 and 100 inclusive are

Their sum is

The numerator is undefined, since 0 raised to the power of 0 is an undefined quantity. Therefore, the entire expression is undefined.

Let's simplify both quantities first before we compare them. becomes because the fractional exponent indicates a square root. We can simplify that by knowing that we can take the square roots of both the numerator and denominator, as shown by: . We can simplify further by taking the square roots (they're perfect squares) and get . Then, let's simplify Column B. To get rid of the negative exponent, we put the numerical expression on the denominator. There's still the fractional exponent at play, so we'll have a square root as well. It looks like this now: . We already simplified , so we can just plug in our answer, , into the denominator. Since we don't want a fraction in the denominator, we can multiply by the reciprocal of , which is 4 to get , which is just 4. Therefore, Column B is greater.

You can simplify Column A first. When you're dividing with exponents and bases are the same, subtract the exponents. Therefore, it simplifies to x. We know that x is positive since it is greater than 1. X is greater than . Try plugging in a number to test. 25 is greater than , which is 5. Even 1.1 is greater than . Therefore, Column A is greater.

The reciprocal of is the quotient of 1 and the number;

This is not in scientific notation, so adjust.

The reciprocal of is the quotient of 1 and the number, or

This is not in scientific notation, so adjust:

For two numbers to be relatively prime, they cannot have any factor in common except for 1. The factors of 18 are 1, 2, 3, 6, 9, and 18.

We can eliminate 32 and 34, since each shares with 18 a factor of 2; we can also eliminate 33 and 39, since each shares with 18 a factor of 3. The factors of 35 are 1, 5, 7, and 35; as can be seen by comparing factors, 18 and 35 only have 1 as a factor, making 35 the correct choice.

To find the prime factorization, break the number down as a product of factors, then keep doing this until all of the factors are prime.

27 has four factors:

Their sum is .

Both are prime factors so this is the prime factorization.

The factors of 30 are 1, 2, 3, 5, 6, 10, 15, and 30. Their sum is

.

40 has as its factors 1, 2, 4, 5, 8, 10, 20, and 40 - a total of eight factors.

(a) 15 has four factors, 1, 3, 5, and 15.

(b) 17, as a prime, has two factors, 1 and 17.

Therefore, (a) is greater.

Each number has only three factors. 121 has 1, 11, and 121 as factors; 169 has 1, 13, and 169 as factors. The answer is that the quantities are equal.

The quanitites are equal, as both can be demonstrated to be equal to .

(a) One of the integers in the given range is , so one of the factors will be , making the product .

(b) The sum of the numbers will be:

(a) The factors of are Their sum is

.

(b) The factors of are Their sum is

.

(b) is greater.

The sum of the integers from to is equal to . We take advantage of the fact that the sum of the even numbers from 10 to 98 is equal to twice the sum of the integers from 5 to 49, as seen here:

First make a factor tree for 16. Keep breaking it down until you get all prime numbers (for example: , which then yields ). Then, at the end, remember to factor the variables as well. Since the b term is squared, that means there are two of them. Therefore, the final answer is .