Exponents - ISEE Upper Level: Quantitative Reasoning

Apply the quotient of powers rule.

In order to add exponential terms, both the base and the exponent must be the same. So we can write:

Based on the zero-exponent rule we have:

That means any non-zero number raised to the zero power is equal to . So we can write:

When exponents are multiplied by one another, and the base is the same, the exponents can be added together.

The first step is to try to create a common base.

Given that the square of 2 is for, the expression can be rewritten as:

When values, having the same base, are multiplied by one another, the exponents are added together and the base stays the same.

Thus,

is equal to

When multiplying exponents, the exponents are added together. Thus, 3 and 7 are added together for a sum of 10. In this problem, the "2" becomes a coefficient in front of the x. Therefore, the correct answer is:

To simplify this problem we need to factor out a

We can do this because multiplying exponents is the same as adding them. Therefore,

When multiplying exponents, the exponents are added together.

When multiplying exponents, the exponents are added together.

It is not necessary to evaluate both terms and multiply.

According to the rules of exponents, when we have the same bases raised to some power that are multiplied with each other, we can add the powers.

This term can be rewritten as a fraction.

The answer is:

Based on the product rule for exponents in order to multiply two exponential terms with the same base, add their exponents:

So we can write:

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 ,

.