What is the Zero Exponent Rule?
Anything (other than 0) to the power of 0 is 1
0^a = 1 for any a ≠ 0.
What does a Negative Exponent indicate?
The expression is in the ‘wrong place’
Negative exponents mean we need to move the base to the opposite side of the fraction.
What is the Product of Powers Rule?
When multiplying exponential numbers with the same bases, add the exponents
a^m * a^n = a^(m+n).
What is the Power to a Power Rule?
When taking an exponential number to another power, multiply the exponents
(a^m)^n = a^(m*n).
What is the Power to a Product Rule?
When taking the product of exponential numbers to another power, multiply the exponents
(a^m * b^n) = a^(mn) * b^(mn).
How do you handle a negative exponent in a fraction?
Put it in a fraction over 1
a^(-n) = 1/(a^n).
What happens if a negative exponent is in the bottom of a fraction?
Move it to the top of the fraction
1/(a^(-n)) = a^n.
What is the Quotient Rule in relation to exponential functions?
When dividing exponential numbers with the same base, we simplify by subtracting the exponents
This rule is essential for simplifying expressions involving exponents.
When taking the quotient of exponential numbers to another power, what do we do?
We simplify by multiplying the exponents
This allows for easier calculations when working with powers of exponential numbers.
What should you remember about negative exponents in your final answer?
Leave no negative exponents
Final answers should be expressed in a standard form without negative exponents.
In the form of y = ab^x, what does ‘a’ represent?
Leading coefficient and initial value (0, a)
‘a’ is crucial for determining the starting point of the exponential function.
What does ‘b’ represent in the exponential function y = ab^x?
Base
The value of ‘b’ determines the growth or decay of the function.
If b > 1, what type of growth does the function exhibit?
Exponential growth
This indicates that the function increases rapidly as x increases.
If 0 < b < 1, what type of growth does the function exhibit?
Exponential decay
This indicates that the function decreases rapidly as x increases.
What is the domain of exponential functions?
All real numbers
Any value for t can be input into the function.
What is the range of exponential functions?
Always positive
The output will never be zero or a negative number.
What is the general form of an exponential growth function?
y = a(1 + r)*
Here, r is the growth rate and a is the initial amount.
What is the general form of an exponential decay function?
y = a(1 - r)*
Here, r is the decay rate and a is the initial amount.
When solving an exponential equation with the same bases, what do we do?
Set the exponents equal and solve
If b^m = b^n, then m = n.
If the bases in an exponential equation are not the same, what can we check?
If we can re-write one or both expressions in the same base
This allows for easier comparison and solving.
What defines a geometric sequence?
Each term is multiplied by a common ratio b
The common ratio is found by dividing the later term by the previous term.
What is the formula for the nth term of a geometric sequence?
a_n = a * b^(n-1)
Here, a is the first term and b is the common ratio.
In a geometric sequence, how do you find the common ratio b?
b = Later term / Previous term
This ratio is constant throughout the sequence.
