Rational exponents and the laws of exponents

Grade 9 Math Worksheet - Building Confidence

Circle the best answer.

1. 1.

   Pia asks why 25^(1/2) equals 5. The law says (a^m)^n = a^(m × n), and Pia squares 25^(1/2). Which reason is correct?

   • A power of 1/2 takes half of the base, so 25^(1/2) = 25 × 1/2 = 12.5.
   • Both 5 and -5 square to 25, so 25^(1/2) is the pair 5 and -5.
   • The 1/2 flips to 2, so 25^(1/2) is the same as 25² = 625.
   • Squaring 25^(1/2) must give 25^(1/2 × 2) = 25^1, and 5² = 25, so 25^(1/2) = 5.
2. 2.

   Ren asks why 8^(1/3) equals 2. The law says (a^m)^n = a^(m × n), and Ren cubes 8^(1/3). Which reason is correct?

   • A power of 1/3 takes a third of the base, so 8^(1/3) = 8 × 1/3 = 8/3.
   • Any fraction power is a square root, so 8^(1/3) is about 2.8.
   • Cubing 8^(1/3) must give 8^(1/3 × 3) = 8^1, and 2³ = 8, so 8^(1/3) = 2.
   • The 1/3 flips to 3, so 8^(1/3) is the same as 8³ = 512.
3. 3.

   Bo asks why 81^(1/2) equals 9. The law says (a^m)^n = a^(m × n), and Bo squares 81^(1/2). Which reason is correct?

   • Squaring 81^(1/2) must give 81^1, and 9² = 81, so 81^(1/2) = 9.
   • Both 9 and -9 square to 81, so 81^(1/2) is the pair 9 and -9.
   • The 1/2 flips to 2, so 81^(1/2) is the same as 81² = 6,561.
   • A power of 1/2 takes half of the base, so 81^(1/2) = 81 × 1/2 = 40.5.
4. 4.

   Milo asks why 27^(1/3) equals 3. The law says (a^m)^n = a^(m × n), and Milo cubes 27^(1/3). Which reason is correct?

   • A power of 1/3 takes a third of the base, so 27^(1/3) = 27 × 1/3 = 9.
   • The 1/3 flips to 3, so 27^(1/3) is the same as 27³ = 19,683.
   • Any fraction power is a square root, so 27^(1/3) is about 5.2.
   • Cubing 27^(1/3) must give 27^1, and 3³ = 27, so 27^(1/3) = 3.
5. 5.

   A cube-shaped box holds 64 cubic inches. Tala says its edge is 64^(1/3) inches. How long is the edge?

   • 4 inches
   • 8 inches
   • about 21.3 inches
   • 3 inches
6. 6.

   Zee's lock code starts with 81^(1/4), one of 4 equal factors of 81. What is 81^(1/4)?

   • 3
   • 20.25
   • 9
   • 4
7. 7.

   Kira's cube-shaped crate holds 125 cubic feet. Its edge is 125^(1/3) feet. How long is the edge?

   • 5 feet
   • 3 feet
   • about 41.7 feet
   • about 11.2 feet
8. 8.

   A cube has edge 5^(1/3) m. Luz multiplies edge × edge × edge for the volume. What is the volume?

   • 5 m³
   • 15 m³
   • 5^(1/27) m³
   • 125 m³
9. 9.

   Jun multiplies four copies of 2^(1/4) together: 2^(1/4) × 2^(1/4) × 2^(1/4) × 2^(1/4). What is the product?

   • 2^(1/256)
   • 2
   • 16
   • 8
10. 10.

    Bo multiplies four copies of 3^(1/2) together. What is the product?

    • 81
    • 12
    • 9
    • 3^(1/16)