Rational or Irrational
Is −0.75 rational or irrational? Explain.
Rational. It terminates and equals −3/4.
Which is greater, √25 or 4.9? Explain.
√25 is greater because √25 = 5 and 5 > 4.9.
Evaluate √144 and ∛(−64).
√144 = 12 and ∛(−64) = −4.
Simplify 2³ · 2⁵. Show the exponent rule step.
2^(3+5) = 2⁸. Add exponents when multiplying the same base.
A student says 12⁰ = 12. Correct the answer and explain.
12⁰ = 1. Any nonzero number to the zero power equals 1.
Is √100 rational or irrational? Explain.
Rational. √100 = 10, which equals 10/1.
Write 23/5 as a decimal, then compare it with 4.65.
23/5 = 4.6, so 23/5 < 4.65.
Is 216 a perfect square, a perfect cube, both, or neither? Explain.
A perfect cube only. 6³ = 216, and 14² = 196 < 216 < 225 = 15².
Simplify 7⁸ / 7⁵. Show the rule step.
7^(8−5) = 7³. Subtract exponents when dividing the same base.
A student says 5⁻² = −25. Correct the answer and explain.
5⁻² = 1/5² = 1/25. A negative exponent means take the reciprocal.
Classify √18 and 0.727272… . Explain.
√18 is irrational because 18 is not a perfect square. 0.727272… is rational because its decimal repeats.
Which two integers is √45 between? Use perfect squares.
6 and 7, because 36 < 45 < 49.
Solve x³ = −216. Check your solution.
x = −6 because (−6)³ = −216.
Simplify (3⁴)². Show the rule step.
3^(4·2) = 3⁸. Multiply exponents for a power of a power.
A student says √36 = ±6. What is the difference between √36 and solving x² = 36?
√36 is the principal square root, 6. The equation x² = 36 has two solutions, 6 and −6.
Write 0.272727… as a fraction. Show an equation.
Let x = 0.272727… . Then 100x = 27.272727… . Subtract: 99x = 27, so x = 27/99 = 3/11.
Estimate √52 to the nearest tenth. Explain why the answer is an approximation.
√52 ≈ 7.2. The decimal does not terminate or repeat, so 7.2 is rounded.
Solve x² = 81. Give ALL real solutions and check each.
x = 9 or −9. Both 9² and (−9)² equal 81.
Simplify a⁵ · a² and a³ / a⁸ using positive exponents. Assume a ≠ 0.
a^(5+2) = a⁷. a^(3−8) = a⁻⁵ = 1/a⁵.
A student writes 3² · 3⁴ = 3⁸. Explain the mistake and correct it.
The student multiplied the exponents. For the same base, add them: 3^(2+4) = 3⁶.
A calculator shows 0.833333333 for 5 ÷ 6. Is the displayed decimal EXACTLY equal to 5/6? Explain.
No. The display is rounded or cut off. The exact decimal is 0.833333… with the 3 repeating forever. 5/6 is rational.
Order least to greatest: √20, 4.6, 9/2, 4.4. Show decimal estimates.
4.4 < √20 < 9/2 < 4.6. Use √20 ≈ 4.472 and 9/2 = 4.5.
Solve x² = 13 exactly. Then find the side lengths of a square with area 100 cm² and a cube with volume 343 cm³.
x = ±√13. Square: s² = 100, s = 10 cm. Cube: s³ = 343, s = 7 cm. Lengths are positive.
Rewrite (3 · 5)² using the power of a product rule. Then simplify (b³)⁴.
(3 · 5)² = 3² · 5² = 225. (b³)⁴ = b^(3·4) = b¹².
Evaluate 6a⁰ + 3a² · b⁻¹ for a = 2 and b = 6. Show substitution.
6(2⁰) + 3(2²)(6⁻¹) = 6(1) + 3(4)(1/6) = 8.