How to divide exponents - ISEE Upper Level Quantitative Reasoning

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 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.

Simplify the following:

Let's begin by recalling that when dividing variables with similar base, we need to subtract the exponents.

To deal with the coefficients, simply treat them as a fraction and simplify.

So our answer is

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.

.