Jeff is working with numbers that are cubed and he is confused. He says that $$ 0^{3}=0 $$ and $$ 2^{3} $$ equals 6, so $$ 5^{3} $$ must be 15 . What does Jeff think the rule for exponents is? What manipulative (a physical object) can you draw to show Jeff a representation of each of the first five cube numbers? Choose the correct answer below. Use the Show Work feature to provide related calculations and additional reasoning to support your answer. A. Jeff is multiplying a number of bases indicated by the exponent, which is the correct procedure. B. Jeff is multiplying the base by the exponent, which is an incorrect procedure. In this case, he is multiplying by three, not taking numbers to the third power. C. Jeff is adding the base and the exponent, which is an incorrect procedure. In this case, he is adding three, not taking numbers to the third power. D. Jeff is adding the product of the base and the exponent to the base, which is an incorrect procedure. In this case, he is adding three times the numbers, not takine numbers to the third power.

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UpStudy AI · Accepted Answer

Jeff’s calculation shows that he is multiplying the base by the exponent. Here’s how we can tell:

1. Jeff calculates $$ 0^3 $$ as:
   $$
   0 \times 3 = 0
   $$
   (This agrees with the actual value $$ 0^3 = 0 $$, but it’s coincidental because multiplying by 0 always gives 0.)

2. For $$ 2^3 $$ he calculates:
   $$
   2 \times 3 = 6
   $$
   However, the correct calculation for $$ 2^3 $$ is:
   $$
   2^3 = 2 \times 2 \times 2 = 8
   $$

3. He then assumes $$ 5^3 $$ is:
   $$
   5 \times 3 = 15
   $$
   The correct value, however, would be:
   $$
   5^3 = 5 \times 5 \times 5 = 125
   $$

Since Jeff’s method consistently follows the rule “multiply the base by 3”, his mistake is that he is multiplying the base by the exponent rather than performing repeated multiplication (exponentiation).

Thus, the correct answer is:

B. Jeff is multiplying the base by the exponent, which is an incorrect procedure. In this case, he is multiplying by three, not taking numbers to the third power.

To help Jeff understand cube numbers, you can use a manipulatives activity with unit cubes. For each cube number, build a larger cube by arranging smaller cubes:

- $$ 0^3 = 0 $$ cubes: no cube.
- $$ 1^3 = 1 $$ cube: a single small cube.
- $$ 2^3 = 8 $$ cubes: arrange 8 small cubes to form a $$ 2 \times 2 \times 2 $$ cube.
- $$ 3^3 = 27 $$ cubes: arrange 27 small cubes to form a $$ 3 \times 3 \times 3 $$ cube.
- $$ 4^3 = 64 $$ cubes: arrange 64 small cubes to form a $$ 4 \times 4 \times 4 $$ cube.

Drawing or physically building these cubes helps visualize that exponentiation represents repeated multiplication (forming a cube in three dimensions) rather than simply multiplying the base by the exponent.
Jeff thinks that to find $$ x^3 $$, you multiply the base by 3. However, the correct method is to multiply the base by itself three times. To show him, use unit cubes to build cubes for each number: 0, 1, 8, 27, and 64. This demonstrates that $$ x^3 $$ means multiplying the base by itself three times, not just multiplying by 3.

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