Is 7 to the negative third power -343?

Answer 1

350

Answer 2

No, but you're close. Negative exponents do not work the same way as negative multiplication. When an exponent is negative, we wind up dividing one by the base. So the answer here would be:

1/343

Sound a bit confusing? Here's an easier way to think about it. Consider what an exponent represents - it's the number of times the base is multiplied by itself. For example, 24 is the same as 2 * 2 * 2 * 2. If we divide that by two, we're left with 2 * 2 * 2, which is the same as 23. We can repeat this, dividing again by 2, which gives us 2 * 2, or 22. This pattern continues. Each time we lower the exponent by 1, we are effectively dividing by the base. Consider then what happens if we keep going:

24 = 2 * 2 * 2 * 2 = 16

23 = 24 / 2 = 2 * 2 * 2 = 8

22 = 23 / 2 = 2 * 2 = 4

21 = 22 / 2 = 2

20 = 21 / 2 = 2 / 2 = 1

2-1 = 20 / 2 = 1/2

2-2 = 2-1 / 2 = 1 / (2 * 2) = 1/4

2-3 = 2-2 / 2 = 1 / (2 * 2 * 2) = 1/8

Notice that there are a few key relationships to be seen here:

Any value raised to a negative exponent is equal to one divided by that same value raised to the positive value of the same exponent:

a-b = 1/(ab)

Anything raised to the power of zero is equal to one (except zero itself, which is a special case):

a0 = 1

Anything raised to the power of 1 is equal to itself:

a1 = a