Question

What is 5 to the power of 3 divided by 5 to the power of 6?

Answer 1

Answer:

  1/5³ = 1/125 = 5⁻³

Step-by-step explanation:

It can be useful to remember that an exponent signifies repeated multiplication.

  5^3 = 5×5×5

  5^6 = 5×5×5×5×5×5

Then the ratio is ...

  $\dfrac{5^3}{5^6}=\dfrac{5\times5\times5}{5\times5\times5\times5\times5\times5}=\dfrac{1}{5\times5\times5}=\boxed{\dfrac{1}{125}}$

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If you want to leave this in terms of exponents, you can see that factors in the denominator cancel (subtract from) those in the numerator. That is ...

  $\dfrac{5^3}{5^6}=\dfrac{5^{3-3}}{5^{6-3}}=\dfrac{5^0}{5^3}=\boxed{\dfrac{1}{5^3}}$

The same sort of exponent arithmetic works to leave a numerator value with a negative exponent:

  $\dfrac{5^3}{5^6}=\dfrac{5^{3-6}}{5^{6-6}}=\dfrac{5^{-3}}{5^0}=\dfrac{5^{-3}}{1}=\boxed{5^{-3}}$

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Additional comment

These ideas are formulated as the rules of exponents:

• (a^b)/(a^c) = a^(b-c)
• 1/(a^b) = a^-b