Define rational and irrational numbers.
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Rational numbers include all integers, decimals, and fractions. Any
terminating or repeating decimal number is a rational number.
Irrational numbers cannot be written as fractions or decimals because the
number of decimal places is infinite and there is no recurring pattern of
digits within the number. For example, pi (π) begins with 3.141592 and
continues without terminating or repeating, so pi is an irrational number.
Explain the laws of exponents.
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, Laws of Exponents.
Any number to the power of 1 is equal to itself: a^1 = a.
Examples: 2^1=2 | -3^1=-3.
The number 1 raised to any power is equal to 1: 1^n = 1.
Examples: 1^3=1 | 1^30=1.
Any number raised to the power of 0 is equal to 1: a^0 = 1.
Examples: 8^0=1 | (-10)^0=1 | (1/2)^0=1.
Add exponents to multiply powers of the same base number: a^n * a^m =
a^(n+m)
Example: 2^3 * 2^4 = 2^(3+4) = 2^7.
Subtract exponents to divide powers of the same base number: a^n/a^m =
a^(n-m)
Example: 2^5/2^3 = 2^(5-3) = 2^2 = 4.
When a power is raised to a power, the exponents are multiplied: (a^n)^m =
a^(n*m)
Example: (3^2)^3 = 3^2 3^2 3^2 = 3^6 = 729.
Multiplication and division operations that are inside parentheses can be
raised to a power. This is the same as each term being raised to that power:
(a b)^n = a^n b^n; (a/b)^n = a^n/b^n;
Multiplication: (2 3)^2 = 2^2 3^2 = 4 * 9 = 36
Division: (4/3)^3 = 4^3/3^3 = 64/27 = 2.37
Explain square roots and perfect squares.
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A root, such as a square root, is another way of writing a fractional
exponent. Instead of using a superscript, roots use the radical symbol (√ )
to indicate the operation. A radical will have a number underneath the bar,
and may sometimes have a number in the upper left: n√a , read as "the nth
root of a." The relationship between radical notation and exponent notation
can be described by this equation: n√a = a^1/n. The two special cases of n =
2 and n = 3 are called square roots and cube roots. If there is no number to
the upper left, it is understood to be a square root (n = 2). Nearly all of the
roots you encounter will be square roots. A square root is the same as a
number raised to the one-half power. When we say that a is the square root
of b (a = √b), we mean that a multiplied by itself equals b: (a × a = b).
A perfect square is a number that has an integer for its square root. There
, are 10 perfect squares from 1 to 100: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100 (the
squares of integers 1 through 10).a
Explain the relationships between percentages, fractions, and decimals.
Visit mometrix.com/academy for related videos.
Enter video codes: 141911, 262335, and 837268
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Percentages can be thought of as fractions that are based on a whole of
100; that is, one whole is equal to 100%. The word percent means per
hundred. Fractions can be expressed as percents by finding equivalent
fractions with a denomination of 100. Example: 7/10 = 70/100 = 70%.
To express a percentage as a fraction, divide the percentage number by
100 and reduce the fraction to its simplest possible terms. Example: 60% =
60/100 = 3/5.
Converting decimals to percentages and percentages to decimals is as
simple as moving the decimal point. To convert from a decimal to a
percent, move the decimal point two places to the right. To convert from a
percent to a decimal, move it two places to the left. Example: 0.23 = 23%;
5.34 = 534%; 0.007 = 0.7%; 700% = 7.00; 86% = 0.86; 0.15% = 0.0015.
It may be helpful to remember that the percentage number will always be
larger than the equivalent decimal number.
Order the following rational numbers from greatest to least: 0.3, 27%, sqrt(100), 72/9,
1/9, 4.5.
Give this one a try later!
Give this one a try later!
Rational numbers include all integers, decimals, and fractions. Any
terminating or repeating decimal number is a rational number.
Irrational numbers cannot be written as fractions or decimals because the
number of decimal places is infinite and there is no recurring pattern of
digits within the number. For example, pi (π) begins with 3.141592 and
continues without terminating or repeating, so pi is an irrational number.
Explain the laws of exponents.
Give this one a try later!
, Laws of Exponents.
Any number to the power of 1 is equal to itself: a^1 = a.
Examples: 2^1=2 | -3^1=-3.
The number 1 raised to any power is equal to 1: 1^n = 1.
Examples: 1^3=1 | 1^30=1.
Any number raised to the power of 0 is equal to 1: a^0 = 1.
Examples: 8^0=1 | (-10)^0=1 | (1/2)^0=1.
Add exponents to multiply powers of the same base number: a^n * a^m =
a^(n+m)
Example: 2^3 * 2^4 = 2^(3+4) = 2^7.
Subtract exponents to divide powers of the same base number: a^n/a^m =
a^(n-m)
Example: 2^5/2^3 = 2^(5-3) = 2^2 = 4.
When a power is raised to a power, the exponents are multiplied: (a^n)^m =
a^(n*m)
Example: (3^2)^3 = 3^2 3^2 3^2 = 3^6 = 729.
Multiplication and division operations that are inside parentheses can be
raised to a power. This is the same as each term being raised to that power:
(a b)^n = a^n b^n; (a/b)^n = a^n/b^n;
Multiplication: (2 3)^2 = 2^2 3^2 = 4 * 9 = 36
Division: (4/3)^3 = 4^3/3^3 = 64/27 = 2.37
Explain square roots and perfect squares.
Give this one a try later!
A root, such as a square root, is another way of writing a fractional
exponent. Instead of using a superscript, roots use the radical symbol (√ )
to indicate the operation. A radical will have a number underneath the bar,
and may sometimes have a number in the upper left: n√a , read as "the nth
root of a." The relationship between radical notation and exponent notation
can be described by this equation: n√a = a^1/n. The two special cases of n =
2 and n = 3 are called square roots and cube roots. If there is no number to
the upper left, it is understood to be a square root (n = 2). Nearly all of the
roots you encounter will be square roots. A square root is the same as a
number raised to the one-half power. When we say that a is the square root
of b (a = √b), we mean that a multiplied by itself equals b: (a × a = b).
A perfect square is a number that has an integer for its square root. There
, are 10 perfect squares from 1 to 100: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100 (the
squares of integers 1 through 10).a
Explain the relationships between percentages, fractions, and decimals.
Visit mometrix.com/academy for related videos.
Enter video codes: 141911, 262335, and 837268
Give this one a try later!
Percentages can be thought of as fractions that are based on a whole of
100; that is, one whole is equal to 100%. The word percent means per
hundred. Fractions can be expressed as percents by finding equivalent
fractions with a denomination of 100. Example: 7/10 = 70/100 = 70%.
To express a percentage as a fraction, divide the percentage number by
100 and reduce the fraction to its simplest possible terms. Example: 60% =
60/100 = 3/5.
Converting decimals to percentages and percentages to decimals is as
simple as moving the decimal point. To convert from a decimal to a
percent, move the decimal point two places to the right. To convert from a
percent to a decimal, move it two places to the left. Example: 0.23 = 23%;
5.34 = 534%; 0.007 = 0.7%; 700% = 7.00; 86% = 0.86; 0.15% = 0.0015.
It may be helpful to remember that the percentage number will always be
larger than the equivalent decimal number.
Order the following rational numbers from greatest to least: 0.3, 27%, sqrt(100), 72/9,
1/9, 4.5.
Give this one a try later!
