Operations with Rational Numbers · Scientific Notation

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Number and Operations · 3 min read

Operations with Rational Numbers · Scientific Notation

Mastery of operations with fractions, decimals, and integers is essential for success throughout the exam.

Adding and Subtracting Fractions

To add or subtract fractions, find a common denominator, convert each fraction, then add or subtract the numerators.

Worked Example

3/4 + 2/5

The least common denominator of 4 and 5 is 20. 3/4 = 15/20 2/5 = 8/20 15/20 + 8/20 = 23/20 = 1 3/20

For mixed numbers, convert to improper fractions or work with whole numbers and fractions separately.

Multiplying Fractions

Multiply numerators together and denominators together. Simplify before or after multiplying.

Worked Example

(3/4) × (2/5) = 6/20 = 3/10

For mixed numbers, convert to improper fractions first.

Dividing Fractions

To divide by a fraction, multiply by its reciprocal.

Worked Example

(3/4) ÷ (2/5) = (3/4) × (5/2) = 15/8 = 1 7/8

Operations with Decimals

Addition and subtraction

Align decimal points and proceed as with whole numbers.

Multiplication

Multiply as with whole numbers, then count total decimal places in both factors and place the decimal in the product accordingly.

Division

When dividing by a decimal, move the decimal point in the divisor to make it a whole number, moving the decimal in the dividend the same number of places.

Operations with Integers

Adding integers

Same signs—add absolute values, keep the sign. Different signs—subtract absolute values, take the sign of the larger absolute value.

Subtracting integers

Add the opposite. For example, 5 − (−3) = 5 + 3 = 8.

Multiplying and dividing integers

Same signs give positive results. Different signs give negative results.

Scientific notation expresses very large or very small numbers in a compact form: a × 10^n, where 1 ≤ |a| < 10 and n is an integer.

Converting to Scientific Notation

Move the decimal point until you have a number between 1 and 10. Count the number of places moved. If you moved the decimal left, the exponent is positive. If you moved it right, the exponent is negative.

Worked Example

45,000,000 = 4.5 × 10^7 (decimal moved 7 places left)
0.00032 = 3.2 × 10^−4 (decimal moved 4 places right)

Converting from Scientific Notation

Move the decimal point the number of places indicated by the exponent. Positive exponents move the decimal right; negative exponents move it left.

Worked Example

6.02 × 10^5 = 602,000
7.1 × 10^−3 = 0.0071

Comparing Numbers in Scientific Notation

First compare exponents. The number with the larger exponent is larger (for positive numbers). If exponents are equal, compare the coefficients.

Worked Example

Compare 3.5 × 10^8 and 9.2 × 10^7
Since 8 > 7, we have 3.5 × 10^8 > 9.2 × 10^7.

Operations with Scientific Notation

Multiplication

Multiply coefficients and add exponents. (3 × 10^4) × (2 × 10^5) = 6 × 10^9

Division

Divide coefficients and subtract exponents. (8 × 10^6) ÷ (2 × 10^2) = 4 × 10^4

Addition and subtraction

Convert to the same power of 10 first, then add or subtract coefficients. (3.5 × 10^4) + (2.1 × 10^3) = (3.5 × 10^4) + (0.21 × 10^4) = 3.71 × 10^4

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← Real Numbers and the Number System · Approximating Irrational Numbers

Operations with Rational Numbers · Scientific Notation

Mastery of operations with fractions, decimals, and integers is essential for success throughout the exam.

Adding and Subtracting Fractions

To add or subtract fractions, find a common denominator, convert each fraction, then add or subtract the numerators.

Worked Example

3/4 + 2/5

The least common denominator of 4 and 5 is 20. 3/4 = 15/20 2/5 = 8/20 15/20 + 8/20 = 23/20 = 1 3/20

For mixed numbers, convert to improper fractions or work with whole numbers and fractions separately.

Multiplying Fractions

Multiply numerators together and denominators together. Simplify before or after multiplying.

Worked Example

(3/4) × (2/5) = 6/20 = 3/10

For mixed numbers, convert to improper fractions first.

Dividing Fractions

To divide by a fraction, multiply by its reciprocal.

Worked Example

(3/4) ÷ (2/5) = (3/4) × (5/2) = 15/8 = 1 7/8

Operations with Decimals

Addition and subtraction

Align decimal points and proceed as with whole numbers.

Multiplication

Multiply as with whole numbers, then count total decimal places in both factors and place the decimal in the product accordingly.

Division

When dividing by a decimal, move the decimal point in the divisor to make it a whole number, moving the decimal in the dividend the same number of places.

Operations with Integers

Adding integers

Same signs—add absolute values, keep the sign. Different signs—subtract absolute values, take the sign of the larger absolute value.

Subtracting integers

Add the opposite. For example, 5 − (−3) = 5 + 3 = 8.

Multiplying and dividing integers

Same signs give positive results. Different signs give negative results.

Scientific notation expresses very large or very small numbers in a compact form: a × 10^n, where 1 ≤ |a| < 10 and n is an integer.

Converting to Scientific Notation

Worked Example

45,000,000 = 4.5 × 10^7 (decimal moved 7 places left)
0.00032 = 3.2 × 10^−4 (decimal moved 4 places right)

Converting from Scientific Notation

Move the decimal point the number of places indicated by the exponent. Positive exponents move the decimal right; negative exponents move it left.

Worked Example

6.02 × 10^5 = 602,000
7.1 × 10^−3 = 0.0071

Comparing Numbers in Scientific Notation

First compare exponents. The number with the larger exponent is larger (for positive numbers). If exponents are equal, compare the coefficients.

Worked Example

Compare 3.5 × 10^8 and 9.2 × 10^7
Since 8 > 7, we have 3.5 × 10^8 > 9.2 × 10^7.

Operations with Scientific Notation

Multiplication

Multiply coefficients and add exponents. (3 × 10^4) × (2 × 10^5) = 6 × 10^9

Division

Divide coefficients and subtract exponents. (8 × 10^6) ÷ (2 × 10^2) = 4 × 10^4

Addition and subtraction

Convert to the same power of 10 first, then add or subtract coefficients. (3.5 × 10^4) + (2.1 × 10^3) = (3.5 × 10^4) + (0.21 × 10^4) = 3.71 × 10^4

Finished reading?

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STAAR Grade 8 Math