Rational exponents and the laws of exponents
Grade 9 Math Worksheet - Challenge Me
Circle the best answer.
A town's population multiplies by 81 over 4 years at a steady yearly factor f. Kira wants f. What is it?
- 21
- 20.25
- 9
- 3
A pond's algae shrinks to 1/27 of its size over 3 days at a steady daily factor f. Zee wants f. What is it?
- 1/3
- about 0.19
- 1/81
- about 0.68
An account multiplies by 25 in one year at a steady rate. Jun wants the factor for each half-year. What is it?
- about 2.9
- 12.5
- 5
- 13
Noah writes 9^(1/2) + 16^(1/2) = 25^(1/2) = 5. Which response is correct?
- No: the exponents add too, so 9^(1/2) + 16^(1/2) = 25^1 = 25.
- No: a half power halves the base, so 9^(1/2) + 16^(1/2) = 4.5 + 8 = 12.5.
- No: 9^(1/2) + 16^(1/2) = 3 + 4 = 7, but 25^(1/2) = 5, so the sum cannot be combined.
- Yes: powers with the same exponent add their bases, so the sum is 25^(1/2) = 5.
Pia writes 4^(1/2) × 9^(1/2) = 36^(1/2) = 6, then checks by finding each root first. Is the step right?
- No: the bases 4 and 9 differ, so the product cannot be rewritten as one power.
- No: multiplying powers adds exponents, so 4^(1/2) × 9^(1/2) = 36^1 = 36.
- No: a half power halves the base, so 4^(1/2) × 9^(1/2) = 2 × 4.5 = 9.
- Yes: 4^(1/2) × 9^(1/2) = 2 × 3 = 6, and 36^(1/2) = 6, so the two roots multiply.
Milo writes (-4)^(1/2) = -2, because the square root of 4 is 2 and the minus sign comes along. Which response is correct?
- No: (-2)² = 4, not -4, and no real number squares to -4.
- Yes: (-2) × (-2) = -4, so -2 squared is -4.
- No: the minus sign drops under a root, so (-4)^(1/2) = 2.
- Yes: the square root of 4 is 2, and the minus sign stays, so it is -2.
Ren solves (x - 2)^(1/2) = 6 for x. What is x?
- about 4.4
- 38
- 36
- 34
Bo solves (x + 5)^(1/3) = 2 for x. What is x?
- 13
- 3
- -1
- 8
Esme solves (x + 3)^(1/2) = 5 for x. What is x?
- 25
- 28
- 22
- 7
Zee solves 2 × x^(1/3) = 8 for x. What is x?
- 4
- 64
- 16
- 512