Exponents - Algebra II

An exponential equation of the form has only one asymptote - a horizontal one at . In the given function, , so its one and only asymptote is .

To find the vertical asymptotes, we set the denominator of the function equal to zero and solve.

For positive values, increases exponentially in the direction and goes to positive infinity, so there is no asymptote on the positive -axis. For negative values, as decreases, the term becomes closer and closer to zero so approaches as we move along the negative axis. As the graph below shows, this is forms a horizontal asymptote.

Factorize both the numerator and denominator.

Notice that one of the binomials will cancel.

The domain of this equation cannot include .

The simplified equation is:

Since the term canceled, the term will have a hole instead of an asymptote.

Set the denominator equal to zero.

Subtract one from both sides.

There will be an asymptote at only:

The answer is:

Factor the numerator and denominator.

Notice that the terms will cancel. The hole will be located at because this is a removable discontinuity.

The denominator cannot be equal to zero. Set the denominator to find the location where the x-variable cannot exist.

The asymptote is located at .

Factor the numerator and denominator.

Rewrite the equation.

Notice that the will cancel. This means that the root of will be a hole instead of an asymptote.

Set the denominator equal to zero and solve for x.

An asymptote is located at:

The answer is:

Begin by distributing the exponent through the parentheses. The power rule dictates that an exponent raised to another exponent means that the two exponents are multiplied:

Any negative exponents can be converted to positive exponents in the denominator of a fraction:

The like terms can be simplified by subtracting the power of the denominator from the power of the numerator:

In order to solve this problem, each of the answer choices needs to be simplified.

Instead of simplifying completely, make all terms into a form such that they have 100 as the exponent. Then they can be easily compared.

, , , and .

Thus, ordering from least to greatest: .

. Thus, is equal to 16.

Step 1: Distribute the exponents in the numberator.

Step 2: Represent the negative exponents in the demoninator.

Step 3: Simplify by combining terms.

When a power applies to an exponent, it acts as a multiplier, so 2a becomes 4a and -b becomes -2b. The negative exponent is moved to the denominator.

Use the power rule to distribute the exponent:

Step 1: Distribute the exponent through the terms in parentheses:

Step 2: Use the division of exponents rule. Subtract the exponents in the numerator from the exponents in the denominator:

When faced with a problem that has an exponent raised to another exponent, the powers are multiplied: then simplify: .

Solve each term separately. A number to the zeroth power is equal to 1, but be careful to apply the signs after the terms have been simplified.

is the correct answer because the order of operations were followed and the multiplication and power rules of exponents were obeyed. These rules are as follows: PEMDAS (parentheses,exponents, multiplication, division, addition, subtraction), for multiplication of exponents follow the format , and .

First we simplify terms within the parenthesis because of the order of operations and the multiplication rule of exponents:

Next we use the power rule to distribute the outer power:

=

**note that in the first step it isn't necessary to combine the two x powers because the individuals terms will still add to x^16 at the end if you use the power rule correctly. However, following the order of operations is a great way to avoid simple math errors and is relevant in many problems.

We begin by distributing the power to all terms within the parentheses. Remember that when we raise a power to a power, we multiply each exponent:

Anytime we have negative exponents, we can convert them to positive exponents. However, if the exponent was negative in the numerator, the term shifts to the denominator. If the exponent was negative in the denominator, the term shifts to the numerator.

To simplify this, we will need to use the power rule and order of operations.

Evaluate the first term. This will be done in two ways to show that the power rule will work for exponents outside of the parenthesis for a single term.

For the second term, we cannot distribute and with the exponent outside the parentheses because it's not a single term. Instead, we must evaluate the terms inside the parentheses first.

Evaluate the second term.

Square the value inside the parentheses.

Subtract the value of the second term with the first term.

First convert into a known base. The number can be rewritten as .

Rewrite the expression.

Use the power rule to multiply the exponents.

Use order of operations to evaluate the expression.