Simplify The Expression:$\[ \frac{8^2}{8^3} - \frac{10^2}{10^3} = \\]

Introduction

In this article, we will simplify the given expression: $\frac{8^2}{8^3} - \frac{10^2}{10^3}$. This expression involves exponents and fractions, and we will use the properties of exponents and fractions to simplify it.

Understanding Exponents

Exponents are a shorthand way of writing repeated multiplication. For example, $a^2$ means $a \times a$, and $a^3$ means $a \times a \times a$. When we have a fraction with exponents, such as $\frac{a^m}{a^n}$, we can simplify it by subtracting the exponents: $\frac{a^m}{a^n} = a^{m-n}$.

Simplifying the First Fraction

Let's start by simplifying the first fraction: $\frac{8^2}{8^3}$. We can use the property of exponents to simplify this fraction. Since the exponent in the numerator is 2 and the exponent in the denominator is 3, we can subtract the exponents: $\frac{8^2}{8^3} = 8^{2-3} = 8^{-1}$.

Simplifying the Second Fraction

Now, let's simplify the second fraction: $\frac{10^2}{10^3}$. Again, we can use the property of exponents to simplify this fraction. Since the exponent in the numerator is 2 and the exponent in the denominator is 3, we can subtract the exponents: $\frac{10^2}{10^3} = 10^{2-3} = 10^{-1}$.

Subtracting the Fractions

Now that we have simplified both fractions, we can subtract them: $\frac{8^2}{8^3} - \frac{10^2}{10^3} = 8^{-1} - 10^{-1}$. To subtract these fractions, we need to have a common denominator. In this case, the common denominator is $8 \times 10 = 80$. So, we can rewrite the fractions as: $\frac{8^{-1}}{1} - \frac{10^{-1}}{1} = \frac{8^{-1}}{1} - \frac{10^{-1}}{1} \times \frac{8}{8} = \frac{8^{-1} \times 8}{1} - \frac{10^{-1} \times 8}{1} = \frac{8^{-1 + 1}}{1} - \frac{10^{-1 + 1}}{1} = \frac{8^0}{1} - \frac{10^0}{1}$.

Simplifying the Result

Now that we have subtracted the fractions, we can simplify the result. Since $8^0 = 1$ and $10^0 = 1$, we can rewrite the result as: $\frac{8^0}{1} - \frac{10^0}{1} = 1 - 1 = 0$.

Conclusion

In this article, we simplified the expression $\frac{8^2}{8^3} - \frac{10^2}{10^3}$. We used the properties of exponents and fractions to simplify the expression, and we arrived at the final result of 0.

Final Answer

The final answer is $\boxed{0}$.

Additional Tips and Tricks

• When simplifying expressions with exponents, make sure to use the properties of exponents to simplify the expression.
• When subtracting fractions, make sure to have a common denominator.
• When simplifying the result, make sure to simplify the expression as much as possible.

Common Mistakes to Avoid

• Not using the properties of exponents to simplify the expression.
• Not having a common denominator when subtracting fractions.
• Not simplifying the result as much as possible.

Real-World Applications

• Simplifying expressions with exponents is an important skill in mathematics, and it has many real-world applications.
• For example, in physics, we often use exponents to describe the behavior of physical systems.
• In engineering, we often use exponents to describe the behavior of complex systems.

Conclusion

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Introduction

In our previous article, we simplified the expression $\frac{8^2}{8^3} - \frac{10^2}{10^3}$. In this article, we will answer some common questions that readers may have about simplifying expressions with exponents.

Q: What is the difference between an exponent and a power?

A: An exponent is a small number that is written above and to the right of a base number. It tells us how many times to multiply the base number by itself. For example, $a^2$ means $a \times a$. A power, on the other hand, is the result of raising a base number to a certain exponent. For example, $a^2$ is a power of $a$.

Q: How do I simplify an expression with exponents?

A: To simplify an expression with exponents, you need to use the properties of exponents. These properties include:

• When multiplying two numbers with the same base, add the exponents.
• When dividing two numbers with the same base, subtract the exponents.
• When raising a number to a power, multiply the exponents.

For example, to simplify the expression $\frac{8^2}{8^3}$, we can use the property of dividing two numbers with the same base: $\frac{8^2}{8^3} = 8^{2-3} = 8^{-1}$.

Q: What is the rule for subtracting fractions with exponents?

A: When subtracting fractions with exponents, you need to have a common denominator. To find the common denominator, multiply the denominators of the two fractions together. Then, rewrite the fractions with the common denominator and subtract them.

For example, to subtract the fractions $\frac{8^2}{8^3}$ and $\frac{10^2}{10^3}$, we can rewrite them as: $\frac{8^{-1}}{1} - \frac{10^{-1}}{1} = \frac{8^{-1} \times 8}{1} - \frac{10^{-1} \times 8}{1} = \frac{8^{-1 + 1}}{1} - \frac{10^{-1 + 1}}{1} = \frac{8^0}{1} - \frac{10^0}{1}$.

Q: How do I simplify a fraction with a negative exponent?

A: To simplify a fraction with a negative exponent, you can rewrite it as a fraction with a positive exponent. To do this, take the reciprocal of the fraction and change the sign of the exponent.

For example, to simplify the fraction $\frac{1}{8^{-1}}$, we can rewrite it as: $\frac{1}{8^{-1}} = \frac{8^1}{1} = 8$.

Q: What is the rule for simplifying expressions with exponents and fractions?

A: To simplify expressions with exponents and fractions, you need to use the properties of exponents and fractions. These properties include:

• When multiplying two numbers with the same base, add the exponents.
• When dividing two numbers with the same base, subtract the exponents.
• When raising a number to a power, multiply the exponents.
• When subtracting fractions, have a common denominator.

For example, to simplify the expression $\frac{8^2}{8^3} - \frac{10^2}{10^3}$, we can use the properties of exponents and fractions: $\frac{8^2}{8^3} - \frac{10^2}{10^3} = 8^{-1} - 10^{-1} = \frac{8^{-1}}{1} - \frac{10^{-1}}{1} = \frac{8^{-1} \times 8}{1} - \frac{10^{-1} \times 8}{1} = \frac{8^{-1 + 1}}{1} - \frac{10^{-1 + 1}}{1} = \frac{8^0}{1} - \frac{10^0}{1}$.

Conclusion

In this article, we answered some common questions that readers may have about simplifying expressions with exponents. We covered topics such as the difference between an exponent and a power, simplifying expressions with exponents, subtracting fractions with exponents, and simplifying expressions with exponents and fractions. By following the rules and properties of exponents and fractions, you can simplify complex expressions and arrive at the final result.

Final Answer

The final answer is $\boxed{0}$.

Additional Tips and Tricks

• When simplifying expressions with exponents, make sure to use the properties of exponents to simplify the expression.
• When subtracting fractions, make sure to have a common denominator.
• When simplifying the result, make sure to simplify the expression as much as possible.

Common Mistakes to Avoid

• Not using the properties of exponents to simplify the expression.
• Not having a common denominator when subtracting fractions.
• Not simplifying the result as much as possible.

Real-World Applications

• Simplifying expressions with exponents is an important skill in mathematics, and it has many real-world applications.
• For example, in physics, we often use exponents to describe the behavior of physical systems.
• In engineering, we often use exponents to describe the behavior of complex systems.