Rational Numbers
Which TWO expressions solve x³ = 125:
5, −5, √125, ∛125?
Is the solution rational or irrational?
5 and ∛125. The solution is rational because ∛125 = 5
How many of these numbers are between 3 and 4? Identify them: √11, ∛8, √15, ∛125, √20.
√11 and √15.
What is ∛216? Show a multiplication equation that proves your answer.
6, because 6 × 6 × 6 = 216.
Maya says 5⁶ ÷ 5⁻² = 5⁴ because 6 − 2 = 4. Is she correct? Explain and give the correct answer.
No. Subtract the entire exponent: 6 − (−2) = 8. The correct answer is 5⁸.
Which expressions are correctly written in scientific notation:
4.8 × 10⁵
48 × 10⁴
0.48 × 10⁶
7.2 × 10⁻³
4.8 × 10⁵ and 7.2 × 10⁻³
What is a rational number?
A number that can be written as a fraction of two integers, with a denominator that is not zero.
A student says, “Every decimal that never ends is irrational.” Is the student correct?
No. A decimal that never ends but repeats is rational. Irrational decimals never end and never repeat.
Find both solutions to x² = 7. Write your answer using the ± symbol.
x = ±√7.
What is the product rule for exponents?
When multiplying powers with the same base, keep the base and add the exponents.
How do you convert a number from scientific notation to standard form? Then convert 6.3 × 10⁻⁴.
Move the decimal right for a positive exponent and left for a negative exponent. The exponent tells how many places to move.
6.3 × 10⁻⁴ = 0.00063.
Give an example of a nonterminating decimal that is rational.
0.333… It is rational because the digit 3 repeats forever.
What is an irrational number?
A number that cannot be written as a fraction of two integers. Its decimal never ends and never repeats.
What is a square root?
A number that, when multiplied by itself, equals the original number.
2³ × 4ⁿ = 2¹³
Two methods:
divide 2³ from both sides
evaluate the powers you know
repeated multiplication or repeated division
n=5
What is scientific notation?
A way to write a number as a coefficient multiplied by a power of 10. For positive numbers, the coefficient is at least 1 but less than 10, and the exponent is an integer
Is 0.272727… a terminating or repeating decimal?
Repeating. The digits 27 repeat forever.
Identify all the irrational numbers:
√49, 3π, 2.75, √18, −0.6, ∛27.
3π and √18.
What is a cube root?
A number that, when used as a factor three times, equals the original number.
Find n: (3ⁿ)⁵ = 3²⁰.
n = 4, because 5n = 20
A video has 78,600,000 views. Write this number in scientific notation, rounding the coefficient to the nearest whole number.
8 × 10⁷.
What is a terminating decimal? Give an example.
A decimal that ends. Examples include 0.75, 1.2, or 0.625.
Is √30 rational or irrational? Explain.
Irrational, because 30 is not a perfect square.
What is √64? Show a multiplication equation that proves your answer.
8, because 8 × 8 = 64.
What is the quotient rule for exponents?
When dividing powers with the same nonzero base, keep the base and subtract the denominator’s exponent from the numerator’s exponent.
How do you convert a positive number from standard form to scientific notation? Then convert 920,000.
Move the decimal until the coefficient is at least 1 but less than 10. Count the places moved to find the exponent. Moving left gives a positive exponent; moving right gives a negative exponent.
920,000 = 9.2 × 10⁵.