How to divide exponents - ISEE Upper Level Quantitative Reasoning

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:

Apply the quotient of powers rule.

Separate the fraction and apply the quotient of powers rule:

Any nonzero number raised to the power of 0 is equal to 1, so , and

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

However, an expression with a denominator of 0 is undefined, so that is the correct choice.

Based on the quotient rule for exponents, we know that for any non-zero number and any integers and ,

.

We should separate the fraction and apply the quotient rule for exponents:

Based on the quotient rule for exponents, we know that for any non-zero number and any integers and ,

.

We also know that any non-zero number raised to the power of is .

We know that any non-zero number raised to the power of is equal to .

Now factor:

We know that any non-zero number raised to the power of is equal to .

Based on the quotient rule for exponents, we know that, for any non-zero number and any integers and ,

.

When a fraction is raised to a power, the numerator and the denominator are both raised to that power:

The power rule says that in order to raise a power to a power, we need to multiply the exponents, i.e.

,

where and are real numbers.

When a fraction is raised to a power, the numerator and the denominator are both raised to that power:

The power rule says that in order to raise a power to a power, we need to multiply the exponents, i.e.

.

The Negative Exponent Rule says .

We know that any non-zero number raised to the power of is equal to .

When a fraction is raised to a power, the numerator and the denominator are both raised to that power. In addition, the power rule says that in order to raise a power to a power, multiply the exponents, i.e.

.

Based on the negative exponent rule we have:

Which says negative exponents in the numerator get moved to the denominator and become positive exponents. And negative exponents in the denominator get moved to the numerator and become positive exponents. So we can write:

Based on the power rule for exponents we can write:

That means; to raise a power to a power we need to multiply the exponents.So we have:

Separate the equation: