Questions and Answers
Anything to the zero power equals 0.
False
What does $a^m * a^n$ equal?
a^{m+n}
What is the result of $(ab)^n$?
a^n * b^n
What does $(a^n)^m$ equal?
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What is the result of $a^m / a^n$?
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What does $(a/b)^n$ equal?
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What does $a^{-n}$ equal?
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What does $x^{a/b}$ equal?
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What is a root?
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What are the fourth roots of 16?
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Does -16 have real fourth roots?
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Who is the only real cube root of -125?
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Odd roots of negatives are acceptable.
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Even roots of negatives are acceptable.
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When taking an even root, what should you use if you don't know whether it's positive?
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In the expression $n√a$, $n$ is the ______ and $a$ is the ______.
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What is the principal root?
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When a is even, what does $n√a^n$ equal?
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What is the result of multiplying two radicals $n√a * n√b$?
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What are like radicals?
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What should you use when asked to multiply functions?
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Radicals can also be expressed as what?
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What is $3√x^2$?
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What is $x^{5/4}$?
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What does the index of a radical become?
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What is often easier when simplifying radicals?
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What is the negative exponent review?
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What is a radical equation?
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How do you solve radical equations?
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How can you solve equations of the form $x^{m/n} = k$?
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What is the addition function operation?
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What is the subtraction function operation?
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What is the multiplication function operation?
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What is the division function operation?
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What does $(F g)(x)$ equal?
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What does $(G f)(x)$ equal?
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What is an inverse function?
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What do you flip for inverse functions?
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In a function, X's can repeat.
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What happens to ordered pairs for inverse functions?
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How do you find the inverse of a function?
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What is the composition of inverse functions?
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What are the key points for square root radical functions?
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What are the key points for cube root radical functions?
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What should you do to make an inverse a function?
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What does H signify in function transformations?
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What does K signify in function transformations?
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What does A signify in function transformations?
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Study Notes
Exponential and Radical Properties
- Any number to the zero power equals one; specifically, ( a^0 = 1 ) (where ( a \neq 0 )).
- Multiplying powers with the same base involves adding exponents: ( a^m \cdot a^n = a^{m+n} ).
- Exponent distribution through parentheses: ( (ab)^n = a^n \cdot b^n ).
- Raising a power to another power means multiplying the exponents: ( (a^n)^m = a^{n \cdot m} ).
- Dividing powers with the same base means subtracting the exponents: ( \frac{a^m}{a^n} = a^{m-n} ).
- Exponent distribution to fractions: ( \left(\frac{a}{b}\right)^n = \frac{a^n}{b^n} ).
- Negative exponents convert to reciprocals: ( a^{-n} = \frac{1}{a^n} ).
Roots and Their Properties
- For any real numbers ( a ) and ( b ), if ( a^n = b ), then ( a ) is the nth root of ( b ).
- Fourth roots of 16: Both ( 2 ) and ( -2 ) are valid fourth roots since ( 2^4 = 16 ) and ( (-2)^4 = 16 ).
- Negative numbers do not have even roots among real numbers (e.g., no real solutions for ( x^4 = -16 )).
- Unique cube root: Only ( -5 ) is the real cube root of ( -125 ).
- Odd roots of negative numbers are valid, while even roots are not.
- For even roots, use the absolute value if the value's positivity is unknown.
Radicals and Rational Exponents
- Notation: ( n\sqrt{a} ) signifies that ( n ) is the index and ( a ) is the radicand.
- The principal root is the positive root when two real roots exist; indicated by the radicand’s sign.
- Negative numbers under even roots yield ( n\sqrt{a^n} = |a| ).
- Multiplying radicals: If ( n\sqrt{a} ) and ( n\sqrt{b} ) are real, then ( n\sqrt{a} \cdot n\sqrt{b} = n\sqrt{ab} ).
- Expressions with like radicals must share the same index and radicand for addition.
Functions and Their Operations
- Function addition: ( (F + g)(x) = f(x) + g(x) ).
- Function subtraction: ( (F - g)(x) = f(x) - g(x) ).
- Function multiplication: ( (F \cdot g)(x) = f(x) \cdot g(x) ).
- Function division requires ensuring ( g(x) \neq 0 ): ( (F/g)(x) = \frac{f(x)}{g(x)} ).
- Composition of functions: ( (F \circ g)(x) = f(g(x)) ).
Inverse Functions
- An inverse function effectively reverses the original function’s operations.
- Inverse functions switch the domain and range.
- To find an inverse, rewrite the function with x and y, switch variables, and solve for ( y ).
- For inverse function composition, ( (f \circ f^{-1})(x) = x ).
Radical Functions
- Square root function: ( y = \sqrt{x} ) with key points ( (0,0), (1,1), (4,2), (9,3) ).
- Cube root function: ( y = \sqrt[3]{x} ) with key points ( (0,0), (1,1), (8,2) ).
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Transformations in functions include:
- Horizontal translations represented by ( h ).
- Vertical translations denoted by ( k ).
- Vertical stretches or shrinks determined by ( a ), where negative ( a ) reflects the graph.
Simplifying Expressions
-
Rational exponents are often simpler for simplifying radicals:
- ( 3\sqrt{x^2} = x^{2/3} ).
- ( x^{5/4} = \sqrt[4]{x^5} ).
- Negative exponent review: Convert ( a^{-n} ) to ( \frac{1}{a^n} ) and ( \frac{1}{a^{-n}} ) to ( a^n ).
Radical Equations
- Radical equations involve variables under a radical.
- To solve, isolate the radical and raise both sides to the necessary power, checking for extraneous solutions.
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Test your knowledge on exponent rules with this set of flashcards specifically for Algebra 2 Unit 7. Each card presents a key exponent concept, providing definitions and important rules to remember. Master the fundamentals of exponents to excel in your mathematical studies!