How to multiply exponential variables - ISEE Upper Level Quantitative Reasoning

To multiply variables with exponents, add the exponents. With multiple variables, simply add the exponents for each different variable.

Simplified:

To multiply variables with exponents, add the exponents. When there are constants mixed in, multiply the constants separately and put back in the final result:

is the greatest common factor of each term, so distribute it out:

We try to factor by finding two integers with product 4 and sum . However, both of our possible factor pairs fail, since and .

is the complete factorization.

The greatest common factor of the terms in is , so factor that out:

Since all factors here are linear, this is the complete factorization.

This can be achieved by using the pattern of difference of squares:

Applying the binomial square pattern:

Use the distributive property, then collect like terms:

The cube of a sum pattern can be applied here:

The cube of a sum pattern can be applied here:

A trinomial with leading coefficient can be factored by looking for two integers to fill in the boxes:

.

The numbers should have product 20 and sum . However, all of the possible factor pairs fail:

The polynomial is prime.

can be seen to be a perfect square trinomial by taking the square root of the first and last terms, multiplying their product by 2, then comparing it to the second term:

Therefore,

To obtain the constant term of a perfect square trinomial, divide the linear coefficient, which here is 20, by 2, and square the quotient. The result is

To obtain the constant term of a perfect square trinomial, divide the linear coefficient, which here is 9, by 2, and square the quotient. The result is

Use the pattern, substituting .

First, simplify all of the exponents. When the exponent is outside of the parantheses, multiply it by the exponents inside so that you get: . Multiply so that you get 27. Then, multiply like terms. First, multilpy 2 by 27 so that you get 54. Then, multiply the x terms. Remember, when bases are the same, add the exponents: . Then, multiply the y terms: . Then, multiply all of the terms together so that you get .

Simplify the following expression:

To combine these, we need to multiply our coefficients and our variables.

First, multiply the coefficients

Next, multiply our variables by adding the exponent:

So, we put it all together to get:

When multiply variables with exponents, we will use the following formula:

So, we can write the problem like this: