​THE MEANING OF DECIMAL EXPONENTS  

1. Express $0.7$ as a fraction, and rewrite $10^{0.7}$ using this fraction.

   Hint (a):

   $0.7=\frac{7}{10}$

2. The power property of exponents can be used to break up this fraction into two factors. Write the value of $c$ so that $10^{0.7}=(10^c)^7$.

   Hint (b):

   Solve: $7c=0.7$

3. Rewrite your answer to part (b) as a root of $10$ raised to a certain power by copying and filling in the blanks of $( ^\square\sqrt { 10 } ) \square$.

   Hint (c):

   $(\sqrt[c]{10})^7$

4. Why is it generally better to take the root first, especially when you are working without a calculator?

   Answer (d):

   The numbers remain smaller and therefore easier to work with.

5. Use a calculator to evaluate your expression from part (c).

6. Now calculate $10^{0.7}$. How does this answer compare with the previous one?

7. Reshma notices that the answer for $10^{0.7}$ is close to $5$. Kahlil knows she can get a value closer to $5$ by using more decimal places in the exponent. Use guess and check to determine $p$ (to the nearest $0.001$) so that $10^p$ is as close to $5$ as possible.

   Hint (g):

   Since $10^{0.7}$ is slightly larger than $5$, choose an exponent a little less than $0.7$.

8. In a flash of brilliance, Reshma suddenly knows how to get several more decimal places instantly. What keys can she press on her calculator to do this?

   Hint (h):

   Rewrite $10^x=5$ using a logarithm.