THE MEANING OF DECIMAL EXPONENTS

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

   Answer (a):

   $10^\frac{7}{10}$

2. We can use the power law for exponents to break up this fraction into two factors. Find the value of $c$ so that $10^{0.7} = \left(10^{c}\right)^{7}$.

   Answer (b):

   $c = \frac{1}{10}$

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

   Answer (c):

   $(10^\frac{1}{10})^7 = (\sqrt[10]{10})^7$

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

   Answer (d):

   The numbers remain smaller and therefore easier to work with.

5. Do the calculation on your graphing calculator.

   Answer (e):

   $10^{0.7} \approx 5.012$

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

   Answer (f):

   $(10^\frac{1}{10})^7 = (\sqrt[10]{10})^7 \approx 5.012$

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

   Answer (g):

   $10^{0.699} \approx 5.000 \text{ is accurate to }0.001.$

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?

   Answer (h):

   She can rewrite $10^{x} = 5$ as $\log_{10}\left(5\right)=x$, then calculate $\log\left(5\right)=0.69897...$ gives several more decimal places instantly.