Rewrite The Following Expression So That It Makes Sense:$1.2 \times 0.1 \quad 9^8 \times 9^3$

Introduction

In mathematics, simplifying expressions is an essential skill that helps us solve problems efficiently and accurately. When dealing with exponential expressions, it's crucial to understand the rules of exponents to simplify them correctly. In this article, we'll focus on rewriting the expression $1.2 \times 0.1 \quad 9^8 \times 9^3$ in a way that makes sense.

Understanding Exponents

Before we dive into simplifying the expression, let's review the basics of exponents. An exponent is a small number that is placed above and to the right of a base number. It tells us how many times to multiply the base number by itself. For example, $2^3$ means $2$ multiplied by itself $3$ times, which equals $8$.

Simplifying the Expression

Now, let's simplify the expression $1.2 \times 0.1 \quad 9^8 \times 9^3$. To do this, we need to follow the order of operations (PEMDAS):

1. Parentheses: There are no parentheses in the expression, so we move on to the next step.
2. Exponents: We can simplify the exponents by adding their exponents. According to the rule of exponents, when we multiply two numbers with the same base, we add their exponents. In this case, we have $9^8 \times 9^3$, which can be simplified to $9^{8+3} = 9^{11}$.
3. Multiplication: Now that we have simplified the exponents, we can multiply the numbers. However, we need to convert the decimal numbers to fractions to make the multiplication easier. $1.2$ can be written as $\frac{12}{10}$, and $0.1$ can be written as $\frac{1}{10}$.
4. Simplifying the Fractions: Now that we have the fractions, we can simplify them by finding the greatest common divisor (GCD) of the numerator and denominator. In this case, the GCD of $12$ and $10$ is $2$, so we can simplify the fraction $\frac{12}{10}$ to $\frac{6}{5}$.
5. Multiplying the Fractions: Now that we have simplified the fractions, we can multiply them. $\frac{6}{5} \times \frac{1}{10} = \frac{6}{50} = \frac{3}{25}$.
6. Multiplying the Exponential Expression: Now that we have simplified the fractions, we can multiply the exponential expression. $9^{11} \times \frac{3}{25} = \frac{9^{11} \times 3}{25}$.

Conclusion

In conclusion, rewriting the expression $1.2 \times 0.1 \quad 9^8 \times 9^3$ in a way that makes sense requires us to follow the order of operations (PEMDAS) and simplify the exponents, fractions, and exponential expression. By doing so, we can simplify the expression to $\frac{9^{11} \times 3}{25}$.

Step-by-Step Solution

Here's a step-by-step solution to the problem:

1. Simplify the exponents: $9^8 \times 9^3 = 9^{8+3} = 9^{11}$
2. Convert the decimal numbers to fractions: $1.2 = \frac{12}{10}$ and $0.1 = \frac{1}{10}$
3. Simplify the fractions: $\frac{12}{10} = \frac{6}{5}$
4. Multiply the fractions: $\frac{6}{5} \times \frac{1}{10} = \frac{6}{50} = \frac{3}{25}$
5. Multiply the exponential expression: $9^{11} \times \frac{3}{25} = \frac{9^{11} \times 3}{25}$

Final Answer

The final answer is $\boxed{\frac{9^{11} \times 3}{25}}$.

Common Mistakes

When simplifying exponential expressions, it's easy to make mistakes. Here are some common mistakes to avoid:

• Not following the order of operations (PEMDAS)
• Not simplifying the exponents correctly
• Not converting decimal numbers to fractions
• Not simplifying the fractions correctly
• Not multiplying the exponential expression correctly

Tips and Tricks

Here are some tips and tricks to help you simplify exponential expressions:

• Always follow the order of operations (PEMDAS)
• Simplify the exponents first
• Convert decimal numbers to fractions
• Simplify the fractions before multiplying
• Multiply the exponential expression last

Practice Problems

Here are some practice problems to help you practice simplifying exponential expressions:

1. Simplify the expression $2^5 \times 2^3$
2. Simplify the expression $3^4 \times 3^2$
3. Simplify the expression $4^3 \times 4^2$
4. Simplify the expression $5^2 \times 5^4$
5. Simplify the expression $6^3 \times 6^5$

Conclusion

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Introduction

In our previous article, we discussed how to simplify exponential expressions by following the order of operations (PEMDAS) and simplifying the exponents, fractions, and exponential expression. However, we know that practice makes perfect, and the best way to learn is by asking questions and getting answers. In this article, we'll provide a Q&A guide to help you understand and simplify exponential expressions.

Q: What is the order of operations (PEMDAS)?

A: The order of operations (PEMDAS) is a set of rules that tells us which operations to perform first when we have multiple operations in an expression. The acronym PEMDAS stands for:

• Parentheses: Evaluate expressions inside parentheses first.
• Exponents: Evaluate any exponential expressions next.
• Multiplication and Division: Evaluate any multiplication and division operations from left to right.
• Addition and Subtraction: Finally, evaluate any addition and subtraction operations from left to right.

Q: How do I simplify exponential expressions?

A: To simplify exponential expressions, follow these steps:

1. Simplify the exponents by adding or subtracting their exponents.
2. Convert decimal numbers to fractions.
3. Simplify the fractions by finding the greatest common divisor (GCD) of the numerator and denominator.
4. Multiply the exponential expression by the simplified fraction.

Q: What is the rule for multiplying exponential expressions?

A: When multiplying exponential expressions with the same base, add their exponents. For example, $2^3 \times 2^4 = 2^{3+4} = 2^7$.

Q: How do I simplify expressions with negative exponents?

A: To simplify expressions with negative exponents, follow these steps:

1. Rewrite the negative exponent as a positive exponent by inverting the base and changing the sign of the exponent.
2. Simplify the expression using the rules of exponents.

Q: What is the rule for dividing exponential expressions?

A: When dividing exponential expressions with the same base, subtract their exponents. For example, $2^5 \div 2^3 = 2^{5-3} = 2^2$.

Q: How do I simplify expressions with fractional exponents?

A: To simplify expressions with fractional exponents, follow these steps:

1. Rewrite the fractional exponent as a product of a power and a root.
2. Simplify the expression using the rules of exponents.

Q: What are some common mistakes to avoid when simplifying exponential expressions?

A: Some common mistakes to avoid when simplifying exponential expressions include:

• Not following the order of operations (PEMDAS)
• Not simplifying the exponents correctly
• Not converting decimal numbers to fractions
• Not simplifying the fractions correctly
• Not multiplying the exponential expression correctly

Q: How can I practice simplifying exponential expressions?

A: You can practice simplifying exponential expressions by working through practice problems, such as:

• Simplifying expressions with the same base
• Simplifying expressions with different bases
• Simplifying expressions with negative exponents
• Simplifying expressions with fractional exponents

Conclusion

In conclusion, simplifying exponential expressions requires us to follow the order of operations (PEMDAS) and simplify the exponents, fractions, and exponential expression. By asking questions and getting answers, we can better understand and simplify exponential expressions. Remember to always follow the order of operations, simplify the exponents first, convert decimal numbers to fractions, simplify the fractions before multiplying, and multiply the exponential expression last. With practice, you'll become proficient in simplifying exponential expressions.

Practice Problems

Here are some practice problems to help you practice simplifying exponential expressions:

1. Simplify the expression $2^5 \times 2^3$
2. Simplify the expression $3^4 \times 3^2$
3. Simplify the expression $4^3 \times 4^2$
4. Simplify the expression $5^2 \times 5^4$
5. Simplify the expression $6^3 \times 6^5$

Answer Key

Here are the answers to the practice problems:

1. $2^{5+3} = 2^8$
2. $3^{4+2} = 3^6$
3. $4^{3+2} = 4^5$
4. $5^{2+4} = 5^6$
5. $6^{3+5} = 6^8$

Common Mistakes

Here are some common mistakes to avoid when simplifying exponential expressions:

• Not following the order of operations (PEMDAS)
• Not simplifying the exponents correctly
• Not converting decimal numbers to fractions
• Not simplifying the fractions correctly
• Not multiplying the exponential expression correctly

Tips and Tricks

Here are some tips and tricks to help you simplify exponential expressions:

• Always follow the order of operations (PEMDAS)
• Simplify the exponents first
• Convert decimal numbers to fractions
• Simplify the fractions before multiplying
• Multiply the exponential expression last

Conclusion

In conclusion, simplifying exponential expressions requires us to follow the order of operations (PEMDAS) and simplify the exponents, fractions, and exponential expression. By asking questions and getting answers, we can better understand and simplify exponential expressions. Remember to always follow the order of operations, simplify the exponents first, convert decimal numbers to fractions, simplify the fractions before multiplying, and multiply the exponential expression last. With practice, you'll become proficient in simplifying exponential expressions.