Rational or Irrational?
Which number must be removed so that every remaining number is rational?
{-7/8, 2.45, sqrt(11), 9/4}
sqrt(11)
11 is not a perfect square. Its square root is irrational; the other numbers can be written as ratios of integers.
Between which two consecutive whole numbers does √58 lie? Explain without rounding first.
7 < sqrt(58) < 8
Because 49 < 58 < 64 and the square roots of 49 and 64 are 7 and 8.
Write an equivalent expression with a positive exponent, then evaluate.
6^-2
6^-2 = 1/6^2 = 1/36
A negative exponent indicates a reciprocal, not a negative value.
Write 0.000684 in scientific notation.
6.84 xx 10^-4
The coefficient is at least 1 and less than 10. Move the decimal four places right to obtain 6.84, so the power of 10 is −4.
Which equation has both 6 and −6 as solutions? Explain.
A. x2=12 B. x3=36 C. x2=36 D. x3=216
C. x2=36
6^2=36
(-6)^2=36
Cubing 6 and −6 produces different values; squaring them gives the same positive value.
Which number is irrational? Explain your choice.
A. √64 B. -3/5 C. √19 D. 0.125
C. √19
19 is not a perfect square. The other values equal 8, −3/5, and 1/8, all rational numbers.
Four points on a number line have the following positions: A = 4.2, B = 4.6, C = 4.9, D = 5.3. Which point best approximates √22? Explain.
Point B at 4.6.
sqrt(22) ~~ 4.6904
Its distance from 4.6 is about 0.09, less than its distance from the other positions.
Simplify. Write your answer as one power of 5 with a positive exponent.
5^-6 xx 5^9
5^(-6+9) = 5^3
Add exponents when multiplying powers with the same base.
A city has 4,783,200 residents. Write an approximation in scientific notation rounded to two significant digits.
4.8 xx 10^6
4,783,200 = 4.7832 × 10⁶; rounding the coefficient to two significant digits gives 4.8.
Solve the equation.
x^3=-343
x=-7
(-7)^3=-343
A negative number has a negative cube root.
Is √72 rational or irrational? Describe its decimal expansion.
Irrational. Its decimal expansion does not terminate and does not eventually repeat, because 72 is not a perfect square.
A point P represents √9 + √13 on a number line. Between which two consecutive integers is P? Estimate its location to the nearest tenth.
sqrt(9) + sqrt(13) ~~ 3 + 3.6056 = 6.6056
Between 6 and 7; approximately 6.6.
Simplify. Write the result using positive exponents.
(7^4 xx 7^-9)^2
(7^-5)^2 = 7^-10 = 1/7^10
First add 4 and −9. Then multiply the resulting exponent by 2.
A laboratory estimates that Sample A contains 2.4 × 105 bacteria and Sample B contains 7.2 × 107 bacteria. How many times as many bacteria are in Sample B as in Sample A?
(7.2 xx 10^7)/(2.4 xx 10^5) = 3 xx 10^2 = 300
Sample B contains 300 times as many bacteria.
A square picture frame encloses an area of 156.25 square inches. What is the perimeter of that square? Show your work.
s=sqrt(156.25)=12.5
Each side is 12.5 inches.
P=4(12.5)=50
The perimeter is 50 inches.
Classify √2.25 as rational or irrational. Explain using its exact value.
sqrt(2.25) = 1.5 = 3/2
Rational, because its value is a ratio of two integers.
Which number is closest to 6.3? Show how you decided.
A. √35 B. √40 C. √43 D. √46
B. √40
Approximate values: 5.92, 6.32, 6.56, and 6.78. The value 6.32 is closest to 6.3.
Evaluate. Give your answer as a simplified fraction.
2^4 xx 9^0 xx 3^-2
16 xx 1 xx (1/9) = 16/9
A nonzero number to the zero power equals 1; a negative exponent gives a reciprocal.
Evaluate and write the result in scientific notation.
(2.1 xx 10^6)/(0.3 xx 10^2)
(2.1/0.3) xx 10^(6-2) = 7 xx 10^4
The result is 70,000. The numerator and denominator need not both be in standard scientific notation to divide them correctly.
A cubic storage bin has a volume of 150 cubic feet. Its side length satisfies s3=150. Between which consecutive whole numbers is s? Estimate s to the nearest tenth of a foot.
Between 5 and 6 feet; about 5.3 feet.
5^3=125 < 150 < 216=6^3
root(3)(150) ~~ 5.3133
Use a cube root, rather than dividing the volume by 3.
A student says, “Any square root containing a decimal is irrational.” Use √0.49 and √0.5 to evaluate the claim.
The claim is false.
sqrt(0.49) = 0.7 = 7/10
This value is rational. √0.5 is irrational: its decimal does not terminate or eventually repeat. A decimal radicand alone does not determine the classification.
Order these real numbers from least to greatest. Show decimal estimates for the irrational numbers.
{-sqrt(7), -2.5, pi, 3 1/5}
-sqrt(7) < -2.5 < pi < 3 1/5
−√7 ≈ −2.646; π ≈ 3.142; 3 1/5 = 3.2. More negative numbers are farther left on the number line.
A student claims these expressions are equivalent. Is the student correct? Simplify each to a single power of 3 and justify your conclusion.
3^8/3^3
3^6 xx 3^-2
No. The first is 35; the second is 34.
3^8/3^3 = 3^(8-3) = 3^5
3^6 xx 3^-2 = 3^(6+(-2)) = 3^4
Subtract exponents when dividing like bases; add them when multiplying like bases.
A sensor records 6.4 × 10-3 meters of motion every second at a constant rate. How much motion does it record in 2.5 × 104 seconds? Give the result in scientific notation and standard form.
(6.4 xx 10^-3)(2.5 xx 10^4) = 16 xx 10^1 = 1.6 xx 10^2
160 meters. Multiply coefficients and add exponents, then rewrite so the coefficient is at least 1 and less than 10.
A square floor mat has area 196 square feet. A cubic box has volume 2,744 cubic feet. Are the mat’s side length and the box’s edge length equal? Find both lengths and justify your answer.
Yes. Both lengths are 14 feet.
sqrt(196)=14
root(3)(2744)=14
The mat uses a square root because area = side². The box uses a cube root because volume = edge³. Physical lengths are positive.