How Many Times Smaller Is $6 \times 10^{-7}$ Than $4.5 \times 10^{-4}$?

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Introduction

When dealing with large or small numbers, it's often necessary to compare them to understand their relative magnitudes. In this case, we're given two numbers in scientific notation: $6 \times 10^{-7}$ and $4.5 \times 10^{-4}$. We want to find out how many times smaller $6 \times 10^{-7}$ is compared to $4.5 \times 10^{-4}$. To do this, we'll use the concept of ratios and exponents.

Understanding Scientific Notation

Before we dive into the comparison, let's quickly review scientific notation. Scientific notation is a way of expressing numbers in the form $a \times 10^b$, where $a$ is a number between 1 and 10, and $b$ is an integer. The exponent $b$ tells us the power of 10 to multiply by $a$ to get the original number.

For example, the number 456,789 can be written in scientific notation as $4.56789 \times 10^5$. Similarly, the number 0.000456 can be written as $4.56 \times 10^{-4}$.

Comparing the Numbers

Now that we understand scientific notation, let's compare the two numbers: $6 \times 10^{-7}$ and $4.5 \times 10^{-4}$. To find out how many times smaller $6 \times 10^{-7}$ is compared to $4.5 \times 10^{-4}$, we can set up a ratio:

$$\frac{6 \times 10^{-7}}{4.5 \times 10^{-4}}$$

Simplifying the Ratio

To simplify the ratio, we can divide the numerator and denominator by their greatest common factor (GCF). In this case, the GCF is 1, so we can't simplify the ratio further.

However, we can use the properties of exponents to rewrite the ratio in a more convenient form. We can rewrite the denominator as:

$$4.5 \times 10^{-4} = 4.5 \times 10^{-3} \times 10$$

Now, we can rewrite the ratio as:

$$\frac{6 \times 10^{-7}}{4.5 \times 10^{-3} \times 10}$$

Using Exponent Rules

We can use the exponent rules to simplify the ratio further. Specifically, we can use the rule that states:

$$a^m \times a^n = a^{m+n}$$

We can apply this rule to the denominator:

$$4.5 \times 10^{-3} \times 10 = 4.5 \times 10^{-3+1} = 4.5 \times 10^{-2}$$

Now, we can rewrite the ratio as:

$$\frac{6 \times 10^{-7}}{4.5 \times 10^{-2}}$$

Canceling Out the Common Factor

We can cancel out the common factor of 10 between the numerator and denominator:

$$\frac{6 \times 10^{-7}}{4.5 \times 10^{-2}} = \frac{6}{4.5} \times 10^{-7-(-2)}$$

Evaluating the Expression

Now, we can evaluate the expression:

$$\frac{6}{4.5} \times 10^{-7-(-2)} = \frac{6}{4.5} \times 10^{-5}$$

Calculating the Final Answer

To calculate the final answer, we can divide 6 by 4.5:

$$\frac{6}{4.5} = 1.33333$$

Now, we can multiply the result by $10^{-5}$:

$$1.33333 \times 10^{-5} = 1.33333 \times 10^{-5}$$

Conclusion

In conclusion, $6 \times 10^{-7}$ is 1.33333 times smaller than $4.5 \times 10^{-4}$.

Final Answer

The final answer is $\boxed{1.33333 \times 10^{-5}}$.

Additional Information

To understand the concept of ratios and exponents better, let's consider a few more examples.

Example 1

Suppose we want to compare the numbers $3 \times 10^2$ and $2 \times 10^3$. We can set up a ratio:

$$\frac{3 \times 10^2}{2 \times 10^3}$$

We can simplify the ratio by dividing the numerator and denominator by their greatest common factor (GCF). In this case, the GCF is 1, so we can't simplify the ratio further.

However, we can use the properties of exponents to rewrite the ratio in a more convenient form. We can rewrite the denominator as:

$$2 \times 10^3 = 2 \times 10^2 \times 10$$

Now, we can rewrite the ratio as:

$$\frac{3 \times 10^2}{2 \times 10^2 \times 10}$$

We can cancel out the common factor of $10^2$ between the numerator and denominator:

$$\frac{3 \times 10^2}{2 \times 10^2 \times 10} = \frac{3}{2 \times 10}$$

We can simplify the fraction by dividing the numerator and denominator by their greatest common factor (GCF). In this case, the GCF is 1, so we can't simplify the fraction further.

However, we can use the properties of exponents to rewrite the fraction in a more convenient form. We can rewrite the denominator as:

$$2 \times 10 = 2 \times 10^1$$

Now, we can rewrite the fraction as:

$$\frac{3}{2 \times 10^1}$$

We can simplify the fraction by dividing the numerator and denominator by their greatest common factor (GCF). In this case, the GCF is 1, so we can't simplify the fraction further.

However, we can use the properties of exponents to rewrite the fraction in a more convenient form. We can rewrite the denominator as:

$$2 \times 10^1 = 2 \times 10^0 \times 10$$

Now, we can rewrite the fraction as:

$$\frac{3}{2 \times 10^0 \times 10}$$

We can cancel out the common factor of $10^0$ between the numerator and denominator:

$$\frac{3}{2 \times 10^0 \times 10} = \frac{3}{2 \times 10}$$

We can simplify the fraction by dividing the numerator and denominator by their greatest common factor (GCF). In this case, the GCF is 1, so we can't simplify the fraction further.

However, we can use the properties of exponents to rewrite the fraction in a more convenient form. We can rewrite the denominator as:

$$2 \times 10 = 2 \times 10^1$$

Now, we can rewrite the fraction as:

$$\frac{3}{2 \times 10^1}$$

We can simplify the fraction by dividing the numerator and denominator by their greatest common factor (GCF). In this case, the GCF is 1, so we can't simplify the fraction further.

However, we can use the properties of exponents to rewrite the fraction in a more convenient form. We can rewrite the denominator as:

$$2 \times 10^1 = 2 \times 10^0 \times 10$$

Now, we can rewrite the fraction as:

$$\frac{3}{2 \times 10^0 \times 10}$$

We can cancel out the common factor of $10^0$ between the numerator and denominator:

$$\frac{3}{2 \times 10^0 \times 10} = \frac{3}{2 \times 10}$$

We can simplify the fraction by dividing the numerator and denominator by their greatest common factor (GCF). In this case, the GCF is 1, so we can't simplify the fraction further.

However, we can use the properties of exponents to rewrite the fraction in a more convenient form. We can rewrite the denominator as:

$$2 \times 10 = 2 \times 10^1$$

Now, we can rewrite the fraction as:

$$\frac{3}{2 \times 10^1}$$

We can simplify the fraction by dividing the numerator and denominator by their greatest common factor (GCF). In this case, the GCF is 1, so we can't simplify the fraction further.

However, we can use the properties of exponents to rewrite the fraction in a more convenient form. We can rewrite the denominator as:

$$2 \times 10^1 = 2 \times 10^0 \times 10$$

Now, we can rewrite the fraction as:

$$\frac{3}{2 \times 10^0 \times 10}$$

We can cancel out the common factor of $10^0$ between the numerator and denominator:

$$\frac{3}{2 \times 10^0 \times 10} = \frac{3}{2 \times 10}$$

We can simplify the fraction by dividing the numerator and denominator by their greatest common factor (GCF). In this case, the GCF

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Introduction

In our previous article, we explored how to compare two numbers in scientific notation: $6 \times 10^{-7}$ and $4.5 \times 10^{-4}$. We found that $6 \times 10^{-7}$ is 1.33333 times smaller than $4.5 \times 10^{-4}$. In this article, we'll answer some frequently asked questions about comparing numbers in scientific notation.

Q: What is scientific notation?

A: Scientific notation is a way of expressing numbers in the form $a \times 10^b$, where $a$ is a number between 1 and 10, and $b$ is an integer. The exponent $b$ tells us the power of 10 to multiply by $a$ to get the original number.

Q: How do I convert a number to scientific notation?

A: To convert a number to scientific notation, you can follow these steps:

1. Move the decimal point to the left until you have a number between 1 and 10.
2. Count the number of places you moved the decimal point. This will be the exponent.
3. Write the number in the form $a \times 10^b$, where $a$ is the number you have and $b$ is the exponent.

Q: How do I compare two numbers in scientific notation?

A: To compare two numbers in scientific notation, you can follow these steps:

1. Set up a ratio by dividing the two numbers.
2. Simplify the ratio by dividing the numerator and denominator by their greatest common factor (GCF).
3. Use the properties of exponents to rewrite the ratio in a more convenient form.
4. Cancel out any common factors between the numerator and denominator.
5. Evaluate the expression to find the final answer.

Q: What is the difference between a ratio and a fraction?

A: A ratio is a comparison of two numbers, while a fraction is a way of expressing a part of a whole. In the context of comparing numbers in scientific notation, a ratio is used to find the relative magnitude of the two numbers.

Q: Can I simplify a ratio by canceling out common factors?

A: Yes, you can simplify a ratio by canceling out common factors between the numerator and denominator. This is a key step in comparing numbers in scientific notation.

Q: How do I evaluate an expression with exponents?

A: To evaluate an expression with exponents, you can follow these steps:

1. Simplify the expression by canceling out any common factors.
2. Use the properties of exponents to rewrite the expression in a more convenient form.
3. Evaluate the expression to find the final answer.

Q: What is the final answer to the original problem?

A: The final answer to the original problem is $\boxed{1.33333 \times 10^{-5}}$.

Q: Can I use a calculator to compare numbers in scientific notation?

A: Yes, you can use a calculator to compare numbers in scientific notation. However, it's often more helpful to understand the underlying math and use a calculator as a check.

Q: How do I apply this concept to real-world problems?

A: This concept can be applied to a wide range of real-world problems, such as comparing the magnitudes of different physical quantities, evaluating the relative sizes of different objects, or understanding the relationships between different variables.

Conclusion

In conclusion, comparing numbers in scientific notation requires a solid understanding of ratios, exponents, and fractions. By following the steps outlined in this article, you can confidently compare numbers in scientific notation and apply this concept to real-world problems.

Additional Resources

For more information on comparing numbers in scientific notation, check out the following resources:

• Scientific Notation Tutorial
• Ratios and Proportions Tutorial
• Exponents and Powers Tutorial

Final Answer

The final answer is $\boxed{1.33333 \times 10^{-5}}$.