Variables - ISEE Upper Level Quantitative Reasoning

By the Binomial Theorem, if is expanded, the coefficient of is

.

Substitute : The coefficient of is:

By the Binomial Theorem, the term of is

.

Substitute and this becomes

.

The coefficient is

.

By the Binomial Theorem, the term of is

,

making the coefficient of

.

We can set in this expression:

Apply the power of a power property twice:

We need to apply the power of power rule twice:

Based on the power of a product rule we have:

The bases are the same, so we can write:

First, recognize that raising the fraction to a negative power is the same as raising the inverted fraction to a positive power.

Apply the exponent within the parentheses and simplify.

This fraction cannot be simplified further.

First, recognize that raising the fraction to a negative power is the same as raising the inverted fraction to a positive power.

Apply the exponent within the parentheses and simplify.

Begin by factoring the numerator and denominator. can be factored out of each term.

can be canceled, since it appears in both the numerator and denomintor.

Next, factor the numerator.

Simplify.

To solve for the variable isolate it on one side of the equation with all of constants on the other side.

First add one third to both sides.

Calculate a common denominator to add the two fractions.

Square both sides to solve for y.

By the Power of a Product Principle,

Also, by the Power of a Power Principle,

Combining these ideas, then substituting:

By the Power of a Power Principle,

So

Also, by the Power of a Product Principle,

, so, substituting,

.

Simplify the following:

Let's recall the rules for distributing exponents.

We treat coefficients (like the 7) like regular numbers and raise them to the new exponent.

We deal with variables (like the t, h, and b) by multiplying their current exponent by the new exponent.

Doing so yields:

Simplify to get:

To divide a variable with an exponent, you can write out all the multiplies, which for this question would look like:

Then cancel out the matching p's, leaving:

Or, if the top and bottom are the same variable, you can simply subtract the bottom exponent from the top exponent.

To divide variables with exponents, either:

Write out the multiplies, then reduce

or subtract the bottom exponent from the top, like so

Remember, negative exponent means dividing. would also be correct, but you should know that is equivalent to

Start by factoring the numerator. can be removed from each term.

Next, expand the denominator.

Simplify by canceling terms.