Algebraic Equations (2)

Question 1

Define exponent.

Answer 1

An exponent indicates how many times a number is multiplied by itself.

Question 2

What is the product of powers rule?

Answer 2

When multiplying like bases, add the exponents: a^m * a^n = a^(m+n).

Question 3

True or false: a^0 equals 1 for any non-zero a.

Answer 3

TRUE

Question 4

Fill in the blank: The power of a power rule states that (a^m)^n = ______.

Answer 4

a^(m*n)

Question 5

What is the quotient of powers rule?

Answer 5

When dividing like bases, subtract the exponents: a^m / a^n = a^(m-n).

Question 6

Define negative exponent.

Answer 6

A negative exponent indicates the reciprocal of the base raised to the opposite positive exponent.

Question 7

Fill in the blank: a^(-n) = ______.

Answer 7

1/a^n

Question 8

What does the power of a product rule state?

Answer 8

To raise a product to a power, raise each factor to that power: (ab)^n = a^n * b^n.

Question 9

True or false: (a/b)^n = a^n / b^n.

Answer 9

TRUE

Question 10

What is the zero exponent rule?

Answer 10

Any non-zero number raised to the power of zero equals one.

Question 11

Fill in the blank: The power of a fraction rule states (a/b)^n = ______.

Answer 11

a^n / b^n

Question 12

Define base in terms of exponents.

Answer 12

The base is the number that is raised to a power in an expression.

Question 13

What is the result of 1^n for any n?

Answer 13

1, because any number raised to any power remains 1.

Question 14

True or false: a^m * a^n = a^(m+n) is an example of the product of powers rule.

Answer 14

TRUE

Question 15

Fill in the blank: The law of exponents helps simplify expressions involving ______.

Answer 15

powers and roots

Question 16

What is the inverse of a positive exponent?

Answer 16

A negative exponent represents the reciprocal of the base raised to the positive exponent.

Question 17

Fill in the blank: a^(m/n) = ______.

Answer 17

n√(a^m)

Question 18

What does the power of a sum rule state?

Answer 18

It states that (a + b)^n cannot be simplified using exponents alone without expansion.

Question 19

True or false: Exponents can only be whole numbers.

Answer 19

FALSE

Exponents can be positive, negative, or fractional.

Question 20

Define fractional exponent.

Answer 20

A fractional exponent indicates both a power and a root: a^(m/n) = n√(a^m).

Question 21

What is the law of exponents?

Answer 21

A set of rules governing the manipulation of exponential expressions.

Question 22

Fill in the blank: The exponent in a^n indicates how many times to multiply ______.

Answer 22

the base a

Question 23

What is the difference of powers rule?

Answer 23

There is no specific rule; however, a^m - a^n cannot be simplified using exponents.

Question 24

True or false: The expression a^m / a^n simplifies to a^(m+n).

Answer 24

FALSE

It simplifies to a^(m-n).

Question 25

Fill in the blank: The **exponential growth** model is represented by the equation y = ______.

Answer 25

a * b^x

Question 26

What is the **exponential decay** model?

Answer 26

A model where quantity decreases at a rate proportional to its current value.

Question 27

Define **exponential function**.

Answer 27

A function of the form f(x) = a * b^x, where b > 0 and b ≠ 1.

Question 28

Fill in the blank: The **base** of an exponential function determines the ______.

Answer 28

rate of growth or decay

Question 29

What is the **relationship** between exponents and logarithms?

Answer 29

Logarithms are the inverse operations of exponentiation.

Question 30

True or false: Logarithms can only be applied to positive numbers.

Answer 30

TRUE

Question 31

Fill in the blank: The **natural logarithm** is denoted as ln and is based on ______.

Answer 31

the constant e

Question 32

What does the **common logarithm** represent?

Answer 32

The logarithm with base 10, denoted as log(x).