Exponents - ISEE Upper Level Quantitative Reasoning

The two quantities are equal.

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.

When you are adding and subtracting terms with exponents, you combine like terms. Since both columns have expressions with the same exponent throughout, you are good to just look at the coefficients. Remember, a coefficient is the number in front of a variable. Therefore, Column A is since . Column B is since . We can see that Column B is greater.

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.

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:

Apply the power of a power property:

Apply the power of a product rule, then apply the power of a power rule:

Any nonzero number raised to the power of 0 is equal to 1.

Use the power of a power property:

(a)

(b)

Since , . Subsequently,

,

making (a) greater

Column A gives simplifies to give us , and Column B simplifies to give us . At first glance, Column B is greater, as it would be for all answers greater than 1. However, if , the two columns are equal. Furthermore, if is negative, or a fraction, Column A is greater. Thus, since we could arrive at all three answers by using different numbers, we cannot determine the answer conclusively.

Anything raised to zero is equal to 1. Therefore, Column A has to be greater because 1 is greater than 0.

Use the difference of squares pattern as follows:

Use the properties of exponents to raise the number to the fourth power:

This is not in scientific notation, so adjust:

is equal to

Move the decimal one place to the right for each number of the exponent with a base ten.

For example, , , etc.

To write 44,000,000 in scientifc notation, write the implied decimal point after the final "0", then move it left until it is after the first nonzero digit (the first "4").

This requires a displacement of seven places, so

, and (B) is greater.