Pre-Algebra Examples

Find the Exponential Function (-3,-8)
Step 1
To find an exponential function, , containing the point, set in the function to the value of the point, and set to the value of the point.
Step 2
Solve the equation for .
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Step 2.1
Rewrite the equation as .
Step 2.2
Rewrite the expression using the negative exponent rule .
Step 2.3
Find the LCD of the terms in the equation.
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Step 2.3.1
Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.
Step 2.3.2
The LCM of one and any expression is the expression.
Step 2.4
Multiply each term in by to eliminate the fractions.
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Step 2.4.1
Multiply each term in by .
Step 2.4.2
Simplify the left side.
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Step 2.4.2.1
Cancel the common factor of .
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Step 2.4.2.1.1
Cancel the common factor.
Step 2.4.2.1.2
Rewrite the expression.
Step 2.5
Solve the equation.
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Step 2.5.1
Rewrite the equation as .
Step 2.5.2
Subtract from both sides of the equation.
Step 2.5.3
Factor the left side of the equation.
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Step 2.5.3.1
Factor out of .
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Step 2.5.3.1.1
Factor out of .
Step 2.5.3.1.2
Rewrite as .
Step 2.5.3.1.3
Factor out of .
Step 2.5.3.2
Rewrite as .
Step 2.5.3.3
Rewrite as .
Step 2.5.3.4
Since both terms are perfect cubes, factor using the sum of cubes formula, where and .
Step 2.5.3.5
Factor.
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Step 2.5.3.5.1
Simplify.
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Step 2.5.3.5.1.1
Apply the product rule to .
Step 2.5.3.5.1.2
Raise to the power of .
Step 2.5.3.5.1.3
Multiply by .
Step 2.5.3.5.1.4
Multiply by .
Step 2.5.3.5.1.5
One to any power is one.
Step 2.5.3.5.2
Remove unnecessary parentheses.
Step 2.5.4
If any individual factor on the left side of the equation is equal to , the entire expression will be equal to .
Step 2.5.5
Set equal to and solve for .
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Step 2.5.5.1
Set equal to .
Step 2.5.5.2
Solve for .
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Step 2.5.5.2.1
Subtract from both sides of the equation.
Step 2.5.5.2.2
Divide each term in by and simplify.
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Step 2.5.5.2.2.1
Divide each term in by .
Step 2.5.5.2.2.2
Simplify the left side.
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Step 2.5.5.2.2.2.1
Cancel the common factor of .
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Step 2.5.5.2.2.2.1.1
Cancel the common factor.
Step 2.5.5.2.2.2.1.2
Divide by .
Step 2.5.5.2.2.3
Simplify the right side.
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Step 2.5.5.2.2.3.1
Move the negative in front of the fraction.
Step 2.5.6
Set equal to and solve for .
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Step 2.5.6.1
Set equal to .
Step 2.5.6.2
Solve for .
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Step 2.5.6.2.1
Use the quadratic formula to find the solutions.
Step 2.5.6.2.2
Substitute the values , , and into the quadratic formula and solve for .
Step 2.5.6.2.3
Simplify.
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Step 2.5.6.2.3.1
Simplify the numerator.
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Step 2.5.6.2.3.1.1
Raise to the power of .
Step 2.5.6.2.3.1.2
Multiply .
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Step 2.5.6.2.3.1.2.1
Multiply by .
Step 2.5.6.2.3.1.2.2
Multiply by .
Step 2.5.6.2.3.1.3
Subtract from .
Step 2.5.6.2.3.1.4
Rewrite as .
Step 2.5.6.2.3.1.5
Rewrite as .
Step 2.5.6.2.3.1.6
Rewrite as .
Step 2.5.6.2.3.1.7
Rewrite as .
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Step 2.5.6.2.3.1.7.1
Factor out of .
Step 2.5.6.2.3.1.7.2
Rewrite as .
Step 2.5.6.2.3.1.8
Pull terms out from under the radical.
Step 2.5.6.2.3.1.9
Move to the left of .
Step 2.5.6.2.3.2
Multiply by .
Step 2.5.6.2.3.3
Simplify .
Step 2.5.6.2.4
Simplify the expression to solve for the portion of the .
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Step 2.5.6.2.4.1
Simplify the numerator.
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Step 2.5.6.2.4.1.1
Raise to the power of .
Step 2.5.6.2.4.1.2
Multiply .
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Step 2.5.6.2.4.1.2.1
Multiply by .
Step 2.5.6.2.4.1.2.2
Multiply by .
Step 2.5.6.2.4.1.3
Subtract from .
Step 2.5.6.2.4.1.4
Rewrite as .
Step 2.5.6.2.4.1.5
Rewrite as .
Step 2.5.6.2.4.1.6
Rewrite as .
Step 2.5.6.2.4.1.7
Rewrite as .
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Step 2.5.6.2.4.1.7.1
Factor out of .
Step 2.5.6.2.4.1.7.2
Rewrite as .
Step 2.5.6.2.4.1.8
Pull terms out from under the radical.
Step 2.5.6.2.4.1.9
Move to the left of .
Step 2.5.6.2.4.2
Multiply by .
Step 2.5.6.2.4.3
Simplify .
Step 2.5.6.2.4.4
Change the to .
Step 2.5.6.2.5
Simplify the expression to solve for the portion of the .
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Step 2.5.6.2.5.1
Simplify the numerator.
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Step 2.5.6.2.5.1.1
Raise to the power of .
Step 2.5.6.2.5.1.2
Multiply .
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Step 2.5.6.2.5.1.2.1
Multiply by .
Step 2.5.6.2.5.1.2.2
Multiply by .
Step 2.5.6.2.5.1.3
Subtract from .
Step 2.5.6.2.5.1.4
Rewrite as .
Step 2.5.6.2.5.1.5
Rewrite as .
Step 2.5.6.2.5.1.6
Rewrite as .
Step 2.5.6.2.5.1.7
Rewrite as .
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Step 2.5.6.2.5.1.7.1
Factor out of .
Step 2.5.6.2.5.1.7.2
Rewrite as .
Step 2.5.6.2.5.1.8
Pull terms out from under the radical.
Step 2.5.6.2.5.1.9
Move to the left of .
Step 2.5.6.2.5.2
Multiply by .
Step 2.5.6.2.5.3
Simplify .
Step 2.5.6.2.5.4
Change the to .
Step 2.5.6.2.6
The final answer is the combination of both solutions.
Step 2.5.7
The final solution is all the values that make true.
Step 2.6
Remove all values that contain imaginary components.
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Step 2.6.1
There are no imaginary components. Add to the final answer.
is a real number
Step 2.6.2
The letter represents an imaginary component, and is not a real number. Do not add to the final answer.
is not a real number
Step 2.6.3
The letter represents an imaginary component, and is not a real number. Do not add to the final answer.
is not a real number
Step 2.6.4
The final answer is the list of values not containing imaginary components.
Step 3
Substitute each value for back into the function to find each possible exponential function.