Excluding Trailing Zeroes, How Many Digits Does $(0.4)^{24}\cdot (0.375)^{22}$ Have To The Right Of The Decimal Point?Hint(s):We Studied Decimal Multiplication By Considering Integers Multiplied By Powers Of 10.

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Introduction

In this article, we will delve into the world of mathematical expressions and explore the concept of excluding trailing zeroes. We will examine the expression (0.4)24⋅(0.375)22(0.4)^{24}\cdot (0.375)^{22} and determine the number of digits it has to the right of the decimal point, excluding trailing zeroes.

Understanding Decimal Multiplication

To tackle this problem, we need to understand how decimal multiplication works. When we multiply two decimal numbers, we can consider the integers multiplied by powers of 10. This concept is crucial in understanding the behavior of decimal numbers.

The Concept of Powers of 10

Powers of 10 are a fundamental concept in mathematics. When we raise 10 to a power, we get a number with a certain number of zeros. For example, 102=10010^2 = 100, which has two zeros. Similarly, 103=100010^3 = 1000, which has three zeros.

Applying the Concept to Decimal Multiplication

When we multiply two decimal numbers, we can consider the integers multiplied by powers of 10. For example, if we multiply 0.4 by 10, we get 4, which is an integer. Similarly, if we multiply 0.375 by 10, we get 3.75, which is also an integer.

**The Expression (0.4)24⋅(0.375)22(0.4)^{24}\cdot (0.375)^{22}

Now, let's apply this concept to the expression (0.4)24⋅(0.375)22(0.4)^{24}\cdot (0.375)^{22}. We can rewrite this expression as (410)24⋅(3751000)22(\frac{4}{10})^{24}\cdot (\frac{375}{1000})^{22}.

Simplifying the Expression

To simplify the expression, we can rewrite it as (410)24⋅(3.7510)22(\frac{4}{10})^{24}\cdot (\frac{3.75}{10})^{22}. Now, we can apply the concept of powers of 10 to simplify the expression further.

Applying the Concept of Powers of 10

When we raise a decimal number to a power, we can consider the integer part multiplied by powers of 10. For example, if we raise 0.4 to the power of 24, we can consider the integer part 4 multiplied by 102410^{24}.

Simplifying the Expression Further

Using this concept, we can simplify the expression further. We can rewrite the expression as (4⋅10−1)24⋅(3.75⋅10−1)22(4\cdot 10^{-1})^{24}\cdot (3.75\cdot 10^{-1})^{22}.

Applying the Concept of Exponents

When we raise a product to a power, we can apply the concept of exponents. For example, if we raise (4⋅10−1)(4\cdot 10^{-1}) to the power of 24, we can apply the exponent 24 to both the integer part 4 and the power of 10.

Simplifying the Expression Even Further

Using this concept, we can simplify the expression even further. We can rewrite the expression as 424⋅(10−1)24⋅3.7522⋅(10−1)224^{24}\cdot (10^{-1})^{24}\cdot 3.75^{22}\cdot (10^{-1})^{22}.

Applying the Concept of Exponents Again

When we raise a product to a power, we can apply the concept of exponents again. For example, if we raise (10−1)24(10^{-1})^{24} to the power of 24, we can apply the exponent 24 to the power of 10.

Simplifying the Expression Once More

Using this concept, we can simplify the expression once more. We can rewrite the expression as 424⋅10−24⋅3.7522⋅10−224^{24}\cdot 10^{-24}\cdot 3.75^{22}\cdot 10^{-22}.

Combining the Terms

Now, we can combine the terms in the expression. We can rewrite the expression as 424⋅3.7522⋅10−24⋅10−224^{24}\cdot 3.75^{22}\cdot 10^{-24}\cdot 10^{-22}.

Simplifying the Expression Even More

Using this concept, we can simplify the expression even more. We can rewrite the expression as 424⋅3.7522⋅10−464^{24}\cdot 3.75^{22}\cdot 10^{-46}.

The Final Expression

Now, we have simplified the expression to its final form. We can see that the expression has a power of 10 with an exponent of -46.

The Number of Digits to the Right of the Decimal Point

To determine the number of digits to the right of the decimal point, we need to consider the exponent of the power of 10. In this case, the exponent is -46.

The Final Answer

Since the exponent is -46, we can conclude that the expression has 46 digits to the right of the decimal point, excluding trailing zeroes.

Conclusion

Introduction

In our previous article, we explored the concept of excluding trailing zeroes and applied it to the expression (0.4)24⋅(0.375)22(0.4)^{24}\cdot (0.375)^{22}. We simplified the expression using the concept of powers of 10 and exponents, and determined that the expression has 46 digits to the right of the decimal point, excluding trailing zeroes. In this article, we will answer some frequently asked questions related to this topic.

Q: What is the concept of excluding trailing zeroes?

A: The concept of excluding trailing zeroes refers to the process of removing the trailing zeroes from a decimal number. This is an important concept in mathematics, especially when dealing with decimal numbers.

Q: How do you exclude trailing zeroes from a decimal number?

A: To exclude trailing zeroes from a decimal number, you can use the concept of powers of 10. For example, if you have a decimal number 0.4, you can multiply it by 10 to get 4, which is an integer. Then, you can raise 10 to a power to get a number with a certain number of zeroes.

Q: What is the significance of powers of 10 in excluding trailing zeroes?

A: Powers of 10 play a crucial role in excluding trailing zeroes. When you raise 10 to a power, you get a number with a certain number of zeroes. This concept is used to simplify decimal numbers and remove trailing zeroes.

Q: How do you apply the concept of exponents to exclude trailing zeroes?

A: To apply the concept of exponents to exclude trailing zeroes, you can raise a product to a power. For example, if you have a decimal number 0.4, you can raise it to the power of 24 to get a number with a certain number of zeroes.

Q: What is the final expression for excluding trailing zeroes?

A: The final expression for excluding trailing zeroes is 424⋅3.7522⋅10−464^{24}\cdot 3.75^{22}\cdot 10^{-46}.

Q: How many digits does the expression have to the right of the decimal point?

A: The expression has 46 digits to the right of the decimal point, excluding trailing zeroes.

Q: What is the significance of the exponent -46 in the expression?

A: The exponent -46 represents the number of zeroes to the right of the decimal point in the expression.

Q: Can you provide an example of excluding trailing zeroes in a real-world scenario?

A: Yes, excluding trailing zeroes is an important concept in finance and economics. For example, when calculating interest rates or investment returns, you need to exclude trailing zeroes to get an accurate result.

Q: How do you apply the concept of excluding trailing zeroes in finance and economics?

A: To apply the concept of excluding trailing zeroes in finance and economics, you can use the concept of powers of 10 and exponents to simplify decimal numbers and remove trailing zeroes.

Conclusion

In this article, we answered some frequently asked questions related to excluding trailing zeroes. We explained the concept of excluding trailing zeroes, applied the concept of powers of 10 and exponents, and determined that the expression has 46 digits to the right of the decimal point, excluding trailing zeroes. We also provided an example of excluding trailing zeroes in a real-world scenario and explained how to apply the concept in finance and economics.