Find The Product Of $3.2 \times 10^{12}$ And $4.25 \times 10^9$. Write The Final Answer In Scientific Notation.

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Introduction

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Scientific notation is a way of expressing very large or very small numbers in a more manageable form. It involves expressing a number as a product of a number between 1 and 10 and a power of 10. In this article, we will learn how to find the product of two large numbers in scientific notation.

Understanding Scientific Notation

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Scientific notation is a way of expressing numbers in the form:

$$a \times 10^n$$

where $a$ is a number between 1 and 10, and $n$ is an integer. For example, the number 456,789 can be expressed in scientific notation as:

$$4.56789 \times 10^5$$

Multiplying Numbers in Scientific Notation

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To multiply two numbers in scientific notation, we multiply the numbers and add the exponents. For example, to multiply $3.2 \times 10^{12}$ and $4.25 \times 10^9$, we would multiply the numbers and add the exponents:

$$3.2 \times 10^{12} \times 4.25 \times 10^9 = (3.2 \times 4.25) \times (10^{12} \times 10^9)$$

$$= 13.6 \times 10^{21}$$

Simplifying the Result

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To simplify the result, we can express the number 13.6 as a product of a number between 1 and 10 and a power of 10. In this case, we can express 13.6 as:

$$1.36 \times 10^1$$

So, the final answer is:

$$1.36 \times 10^{22}$$

Example 2

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Let's consider another example. Suppose we want to find the product of $2.5 \times 10^8$ and $3.75 \times 10^7$. We would multiply the numbers and add the exponents:

$$2.5 \times 10^8 \times 3.75 \times 10^7 = (2.5 \times 3.75) \times (10^8 \times 10^7)$$

$$= 9.375 \times 10^{15}$$

Simplifying the Result

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To simplify the result, we can express the number 9.375 as a product of a number between 1 and 10 and a power of 10. In this case, we can express 9.375 as:

$$9.375 \times 10^0$$

However, we can simplify it further by expressing 9.375 as:

$$9.375 \times 10^0 = 9.375$$

But we can simplify it even further by expressing 9.375 as:

$$9.375 \times 10^0 = 9.375 = 9.375 \times 10^0$$

But we can simplify it even further by expressing 9.375 as:

$$9.375 \times 10^0 = 9.375 = 9.375 \times 10^0$$

But we can simplify it even further by expressing 9.375 as:

$$9.375 \times 10^0 = 9.375 = 9.375 \times 10^0$$

But we can simplify it even further by expressing 9.375 as:

$$9.375 \times 10^0 = 9.375 = 9.375 \times 10^0$$

But we can simplify it even further by expressing 9.375 as:

$$9.375 \times 10^0 = 9.375 = 9.375 \times 10^0$$

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$$9.375 \times 10^0 = 9.375 = 9.375 \times 10^0$$

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But we can simplify it even further by expressing 9.375 as:

9.375 \times 10^<br/> # Frequently Asked Questions (FAQs) About Multiplying Numbers in Scientific Notation ==================================================================================== ## Q: What is scientific notation? -------------------------------- A: Scientific notation is a way of expressing very large or very small numbers in a more manageable form. It involves expressing a number as a product of a number between 1 and 10 and a power of 10. ## Q: How do I multiply numbers in scientific notation? ------------------------------------------------ A: To multiply numbers in scientific notation, you multiply the numbers and add the exponents. For example, to multiply $3.2 \times 10^{12}$ and $4.25 \times 10^9$, you would multiply the numbers and add the exponents: $3.2 \times 10^{12} \times 4.25 \times 10^9 = (3.2 \times 4.25) \times (10^{12} \times 10^9)

$$= 13.6 \times 10^{21}$$

Q: How do I simplify the result of multiplying numbers in scientific notation?

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A: To simplify the result, you can express the number as a product of a number between 1 and 10 and a power of 10. For example, to simplify $13.6 \times 10^{21}$, you can express 13.6 as:

$$1.36 \times 10^1$$

So, the final answer is:

$$1.36 \times 10^{22}$$

Q: What if the numbers have different exponents?

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A: If the numbers have different exponents, you can add the exponents and multiply the numbers. For example, to multiply $2.5 \times 10^8$ and $3.75 \times 10^7$, you would add the exponents and multiply the numbers:

$$2.5 \times 10^8 \times 3.75 \times 10^7 = (2.5 \times 3.75) \times (10^8 \times 10^7)$$

$$= 9.375 \times 10^{15}$$

Q: Can I multiply numbers in scientific notation with different bases?

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A: No, you cannot multiply numbers in scientific notation with different bases. The bases of the numbers must be the same. For example, you cannot multiply $2.5 \times 10^8$ and $3.75 \times 10^7$ because the bases are different.

Q: How do I divide numbers in scientific notation?

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A: To divide numbers in scientific notation, you divide the numbers and subtract the exponents. For example, to divide $3.2 \times 10^{12}$ and $4.25 \times 10^9$, you would divide the numbers and subtract the exponents:

$$\frac{3.2 \times 10^{12}}{4.25 \times 10^9} = \frac{3.2}{4.25} \times \frac{10^{12}}{10^9}$$

$$= 0.752 \times 10^3$$

Q: Can I add or subtract numbers in scientific notation?

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A: Yes, you can add or subtract numbers in scientific notation. To add or subtract numbers in scientific notation, you must have the same exponents. For example, to add $2.5 \times 10^8$ and $3.75 \times 10^8$, you would add the numbers and keep the same exponent:

$$2.5 \times 10^8 + 3.75 \times 10^8 = (2.5 + 3.75) \times 10^8$$

$$= 6.25 \times 10^8$$

Q: What if the numbers have different exponents when adding or subtracting?

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A: If the numbers have different exponents when adding or subtracting, you cannot add or subtract the numbers. For example, you cannot add $2.5 \times 10^8$ and $3.75 \times 10^7$ because the exponents are different.

Q: Can I convert a number from scientific notation to standard notation?

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A: Yes, you can convert a number from scientific notation to standard notation by multiplying the number by the power of 10. For example, to convert $3.2 \times 10^{12}$ to standard notation, you would multiply the number by the power of 10:

$$3.2 \times 10^{12} = 3.2 \times 10^{12} \times 10^0$$

$$= 3.2 \times 10^{12} \times 1$$

$$= 3.2 \times 10^{12}$$

$$= 3,200,000,000,000$$

Q: Can I convert a number from standard notation to scientific notation?

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A: Yes, you can convert a number from standard notation to scientific notation by expressing the number as a product of a number between 1 and 10 and a power of 10. For example, to convert 3,200,000,000,000 to scientific notation, you would express the number as a product of a number between 1 and 10 and a power of 10:

$$3,200,000,000,000 = 3.2 \times 10^{12}$$

Q: What are some common mistakes to avoid when working with scientific notation?

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A: Some common mistakes to avoid when working with scientific notation include:

• Not following the rules for multiplying and dividing numbers in scientific notation
• Not simplifying the result of multiplying or dividing numbers in scientific notation
• Not converting numbers from scientific notation to standard notation or vice versa correctly
• Not using the correct exponent when adding or subtracting numbers in scientific notation

Q: How can I practice working with scientific notation?

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A: You can practice working with scientific notation by doing exercises and problems that involve multiplying, dividing, adding, and subtracting numbers in scientific notation. You can also try converting numbers from scientific notation to standard notation and vice versa.