The Richter scale reading for an earthquake measures its intensity (as a multiple of some minimum intensity used for comparison). The intensity *I* corresponding to a Richter scale reading *R* is given by

*I* = 10*R*.

(a) A quake measuring 6.5 on the Richter scale would be moderate. Express the intensity of such a quake in exponential form and in radical form.


exponential form    

*I*

 = 
 
radical form    

*I*

 = 
 


(b) Find the intensity of a quake measuring 8.


*I* = 



(c) The San Francisco quake that occurred during the 1989 World Series measured 6.9, and the March 2011 quake that devastated Sendai, Japan, measured 9.0. Calculate the ratio of these intensities (larger to smaller). (Round your answer to three decimal places.)

Let's solve each part of the problem step by step.

### (a) Express the intensity of a quake measuring 6.5 on the Richter scale in exponential and radical forms

The intensity \( I \) corresponding to a Richter scale reading \( R \) is given by:

\[
I = 10^R
\]

For a quake measuring 6.5 on the Richter scale:

\[
I = 10^{6.5}
\]

**Exponential form:**

\[
I = 10^{6.5}
\]

**Radical form:**

To express this in radical form, recall that:

\[
10^{6.5} = 10^{\frac{13}{2}} = \sqrt{10^{13}}
\]

So, the radical form is:

\[
I = \sqrt{10^{13}}
\]

### (b) Find the intensity of a quake measuring 8

For a quake measuring 8 on the Richter scale:

\[
I = 10^8
\]

So, the intensity \( I \) is:

\[
I = 100,000,000
\]

### (c) Calculate the ratio of the intensities of the San Francisco quake (6.9) and the Sendai quake (9.0)

The intensities of the quakes are:

For the San Francisco quake:

\[
I_{\text{SF}} = 10^{6.9}
\]

For the Sendai quake:

\[
I_{\text{Sendai}} = 10^{9.0}
\]

The ratio of the intensities (larger to smaller) is:

\[
\text{Ratio} = \frac{I_{\text{Sendai}}}{I_{\text{SF}}} = \frac{10^{9.0}}{10^{6.9}} = 10^{9.0 - 6.9} = 10^{2.1}
\]

Calculate \( 10^{2.1} \):

\[
10^{2.1} \approx 126.491
\]

So, the ratio of the intensities is approximately:

\[
\text{Ratio} \approx 126.491
\]

### Summary of Results:

**(a)**  
- Exponential form: \( I = 10^{6.5} \)  
- Radical form: \( I = \sqrt{10^{13}} \)

**(b)**  
- Intensity for \( R = 8 \): \( I = 100,000,000 \)

**(c)**  
- Ratio of intensities: \( \approx 126.491 \)

Would you like to explore the implications of these results or need further clarification?

---

**Related Questions:**

1. How would you express \( 10^{9.0} \) in radical form?
2. What would be the ratio of intensities between two quakes measuring 7.0 and 8.0 on the Richter scale?
3. If the intensity of a quake doubles, how does the Richter scale reading change?
4. Can you compare the intensity of a quake measuring 5.5 to one measuring 6.0?
5. How does the intensity change if the Richter scale reading increases by 0.5?

**Tip:** Understanding logarithmic scales like the Richter scale is crucial for grasping how small changes in the scale correspond to large differences in intensity.

Explore how Richter scale readings correlate to earthquake intensities through exponential and radical forms. Solve specific examples like finding the intensity for a quake measuring 8 on the Richter scale and calculating intensity ratios between quakes.

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