8TH GRADE Chapter 1 review

What is a real number?

Any number that can be placed on the number line, including both rational and irrational numbers.

Define a rational number.

A number that can be written as a fraction a/b, where a and b are integers and b ≠ 0.

Define an irrational number.

A number that cannot be written as a fraction of two integers. Its decimal expansion is non-repeating and non-terminating (e.g., π, √2).

Is -5 rational or irrational? Why?

Rational, because it can be written as -5 = -5/1.

Is √5 rational or irrational? Why?

Irrational, because its decimal expansion never ends or repeats.

Is 5.262626… rational or irrational?

Rational, because it repeats (can be expressed as a fraction).

What is the closure property?

A set of numbers is closed under an operation if performing the operation on numbers in the set always results in another number in the set.

Are integers closed under addition?

Yes.

Are integers closed under subtraction?

Yes.

Are integers closed under multiplication?

Yes.

Are integers closed under division?

No, because division may produce non-integers.

What is the sum of two rational numbers?

Always rational.

What is the sum of a rational and an irrational number?

Always irrational.

What is the sum of two irrational numbers?

Sometimes rational, sometimes irrational.

True or False: Irrational numbers are closed under multiplication.

False (example: √2 × √2 = 2, which is rational).

True or False: Irrational numbers are closed under subtraction.

False.

What is a radical expression?

An expression that contains a root symbol (√), such as √25.

What is the radicand?

The number inside the radical symbol.

What is the index of a radical?

The small number above and to the left of the radical that indicates the type of root.

What is the default index when none is shown?

2 (square root).

What does a^(1/n) mean?

The nth root of a, or √[n]{a}.

What does a^(m/n) mean?

(√[n]{a})^m.

Simplify 9^(3/2).

(√9)^3 = 3^3 = 27.

Simplify 8^(2/3).

(√[3]{8})^2 = 2^2 = 4.

Simplify 4^(-3/2).

1 / (√4^3) = 1/8.

Simplify (64)^(2/3).

(√[3]{64})^2 = 4^2 = 16.

Simplify (27)^(4/3).

(√[3]{27})^4 = 3^4 = 81.

State the Product of Powers Property.

a^m × a^n = a^(m+n).

State the Quotient of Powers Property.

a^m ÷ a^n = a^(m-n).

State the Power of a Power Property.

(a^m)^n = a^(m × n).

State the Power of a Product Property.

(ab)^n = a^n b^n.

State the Power of a Quotient Property.

(a/b)^n = a^n / b^n.

What is the Zero Exponent Rule?

a^0 = 1 (for a ≠ 0).

What is the Negative Exponent Rule?

a^(-n) = 1 / a^n.

Define accuracy in measurement.

How close a measurement is to the true or accepted value.

Define precision in measurement.

The degree of detail in a measurement, based on the smallest unit used.

True or False: A measurement can be precise but not accurate.

True (results close together but far from the true value).

True or False: A measurement can be accurate but not precise.

True (average close to true value, but results spread out).

What are significant digits?

Digits in a measurement that show precision.

Rule: All nonzero digits are…

Significant.

Rule: Zeros between nonzero digits are…

Significant.

Rule: Leading zeros are…

Not significant.

Rule: Trailing zeros with a decimal point are…

Significant.

Rule: Trailing zeros without a decimal point are…

Not significant.

Example: How many sig figs in 305?

3.

Example: How many sig figs in 0.00045?

2.

Example: How many sig figs in 4.1000?

5.

Multiplication/division rule for sig figs?

Result has the same number of sig figs as the factor with the fewest.

Addition/subtraction rule for sig figs?

Result has the same decimal precision as the least precise measurement.

Accuracy

Closeness of a measurement to the true value.

Closure

Property that a set is closed under an operation if applying the operation stays in the set.

Index

The small number in a radical showing the type of root.

Precision

Level of detail in a measurement.

Radical

A root symbol (√).

Radicand

The number under the radical.

Significant digits

Digits that show precision of a measurement.