Algebra 2 Unit 7 Test: Exponents Flashcards

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?

a^{n*m} Signup and view all the answers

What is the result of $a^m / a^n$?

a^{m-n} Signup and view all the answers

What does $(a/b)^n$ equal?

a^n / b^n Signup and view all the answers

What does $a^{-n}$ equal?

1 / a^n Signup and view all the answers

What does $x^{a/b}$ equal?

b√x^a = (b√x)^a Signup and view all the answers

What is a root?

If $a^n = b$, then $a$ is an nth root of $b$. Signup and view all the answers

What are the fourth roots of 16?

2 and -2 Signup and view all the answers

Does -16 have real fourth roots?

No Signup and view all the answers

Who is the only real cube root of -125?

-5 Signup and view all the answers

Odd roots of negatives are acceptable.

True Signup and view all the answers

Even roots of negatives are acceptable.

False Signup and view all the answers

When taking an even root, what should you use if you don't know whether it's positive?

absolute value Signup and view all the answers

In the expression $n√a$, $n$ is the and $a$ is the .

index, radicand Signup and view all the answers

What is the principal root?

The positive root in cases where a number has two real roots. Signup and view all the answers

When a is even, what does $n√a^n$ equal?

|a| Signup and view all the answers

What is the result of multiplying two radicals $n√a * n√b$?

n√(ab) Signup and view all the answers

What are like radicals?

Radical expressions with the same index and radicand. Signup and view all the answers

What should you use when asked to multiply functions?

distributive property or FOIL Signup and view all the answers

Radicals can also be expressed as what?

rational (or fractional) powers Signup and view all the answers

What is $3√x^2$?

x^{2/3} Signup and view all the answers

What is $x^{5/4}$?

4√x^5 Signup and view all the answers

What does the index of a radical become?

denominator of the rational power Signup and view all the answers

What is often easier when simplifying radicals?

using rational exponents Signup and view all the answers

What is the negative exponent review?

a^{-n} = 1/a^n Signup and view all the answers

What is a radical equation?

An equation with a variable in a radicand or a variable with a radical exponent. Signup and view all the answers

How do you solve radical equations?

Isolate the radical and then raise both sides to the same power. Signup and view all the answers

How can you solve equations of the form $x^{m/n} = k$?

Raise each side to the power $m/n$. Signup and view all the answers

What is the addition function operation?

(F + g)(x) = f(x) + g(x) Signup and view all the answers

What is the subtraction function operation?

(F - g)(x) = f(x) - g(x) Signup and view all the answers

What is the multiplication function operation?

(F * g)(x) = f(x) * g(x) Signup and view all the answers

What is the division function operation?

(F / g)(x) = f(x) / g(x), g(x) cannot = 0 Signup and view all the answers

What does $(F g)(x)$ equal?

f(g(x)) Signup and view all the answers

What does $(G f)(x)$ equal?

g(f(x)) Signup and view all the answers

What is an inverse function?

A function that undoes the process of another function. Signup and view all the answers

What do you flip for inverse functions?

Domain and range Signup and view all the answers

In a function, X's can repeat.

False Signup and view all the answers

What happens to ordered pairs for inverse functions?

You switch the numbers and sides. Signup and view all the answers

How do you find the inverse of a function?

Rewrite with x and y, then switch variables and solve for y. Signup and view all the answers

What is the composition of inverse functions?

(f * f^-1)(x) = x Signup and view all the answers

What are the key points for square root radical functions?

0, 1, 4, 9 Signup and view all the answers

What are the key points for cube root radical functions?

0, 1, 8 Signup and view all the answers

What should you do to make an inverse a function?

Restrict the domain. Signup and view all the answers

What does H signify in function transformations?

Horizontal translation. Signup and view all the answers

What does K signify in function transformations?

Vertical translation. Signup and view all the answers

What does A signify in function transformations?

Vertical stretch or shrink. Signup and view all the answers

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) ).
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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Description

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!