Simplify:(4 -1×2 5)​

Understanding the Problem

When dealing with exponents, it's essential to understand the rules and properties that govern them. In this problem, we're given the expression (4^-1×25) and we need to simplify it. To start, let's break down the components of the expression and understand what each part means.

Exponents and Their Rules

Exponents are a shorthand way of representing repeated multiplication. For example, 2^3 can be read as "2 to the power of 3" and is equivalent to 2 multiplied by itself 3 times: 2 × 2 × 2 = 8. The exponent tells us how many times the base (in this case, 2) is multiplied by itself.

When dealing with negative exponents, we can rewrite them as positive exponents by taking the reciprocal of the base. For example, 4^-1 can be rewritten as 1/4^1. This is because the reciprocal of 4 is 1/4, and the exponent -1 tells us to take the reciprocal of the base.

Simplifying the Expression

Now that we understand the rules of exponents, let's simplify the expression (4^-1×25). We can start by rewriting the negative exponent as a positive exponent:

(4^-1×25) = (1/4^1×25)

Next, we can simplify the expression by combining the bases:

(1/4^1×25) = (1/4×2^5)

Now, we can simplify the expression further by multiplying the bases:

(1/4×2^5) = (1/4×32)

Finally, we can simplify the expression by multiplying the numerators and denominators:

(1/4×32) = 8

Therefore, the simplified expression is 8.

Conclusion

In this problem, we simplified the expression (4^-1×25) by understanding the rules of exponents and applying them to rewrite the negative exponent as a positive exponent. We then combined the bases and multiplied them to simplify the expression further. The final result is 8.

Tips and Tricks

• When dealing with negative exponents, remember to rewrite them as positive exponents by taking the reciprocal of the base.
• When combining bases, make sure to multiply the bases together.
• When multiplying numerators and denominators, make sure to multiply the numerators together and the denominators together.

Common Mistakes

• Forgetting to rewrite negative exponents as positive exponents.
• Not combining bases correctly.
• Not multiplying numerators and denominators correctly.

Real-World Applications

• Exponents are used in many real-world applications, such as finance, science, and engineering.
• Understanding exponents is essential for solving problems in these fields.
• Simplifying expressions with exponents is a crucial skill for anyone working in these fields.

Practice Problems

• Simplify the expression (3^-2×53).
• Simplify the expression (2^4×3-1).
• Simplify the expression (4^2×2-3).

Solutions

• (3^-2×53) = (1/9×125) = 14.44
• (2^4×3-1) = (16×1/3) = 5.33
• (4^2×2-3) = (16×1/8) = 2

Conclusion

In this article, we simplified the expression (4^-1×25) by understanding the rules of exponents and applying them to rewrite the negative exponent as a positive exponent. We then combined the bases and multiplied them to simplify the expression further. The final result is 8. We also provided tips and tricks, common mistakes, real-world applications, and practice problems to help readers understand and apply the concept of exponents.

Frequently Asked Questions

We've received many questions about the expression (4^-1×25) and how to simplify it. Here are some of the most frequently asked questions and their answers:

Q: What is the rule for negative exponents?

A: The rule for negative exponents is to rewrite them as positive exponents by taking the reciprocal of the base. For example, 4^-1 can be rewritten as 1/4^1.

Q: How do I simplify an expression with a negative exponent?

A: To simplify an expression with a negative exponent, follow these steps:

1. Rewrite the negative exponent as a positive exponent by taking the reciprocal of the base.
2. Combine the bases by multiplying them together.
3. Simplify the expression by multiplying the numerators and denominators.

Q: What is the difference between 4^-1 and 1/4?

A: 4^-1 and 1/4 are equivalent expressions. The negative exponent -1 tells us to take the reciprocal of the base, which is 1/4.

Q: Can I simplify an expression with multiple negative exponents?

A: Yes, you can simplify an expression with multiple negative exponents by following the same steps as before:

1. Rewrite each negative exponent as a positive exponent by taking the reciprocal of the base.
2. Combine the bases by multiplying them together.
3. Simplify the expression by multiplying the numerators and denominators.

Q: What is the final result of the expression (4^-1×25)?

A: The final result of the expression (4^-1×25) is 8.

Q: Can I use a calculator to simplify an expression with exponents?

A: Yes, you can use a calculator to simplify an expression with exponents. However, it's always a good idea to understand the rules and properties of exponents before using a calculator.

Q: How do I apply the rules of exponents to real-world problems?

A: The rules of exponents can be applied to many real-world problems, such as finance, science, and engineering. For example, you might need to calculate the interest on a loan or the growth of a population.

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

A: Some common mistakes to avoid when simplifying expressions with exponents include:

• Forgetting to rewrite negative exponents as positive exponents
• Not combining bases correctly
• Not multiplying numerators and denominators correctly

Additional Resources

If you're still having trouble simplifying expressions with exponents, here are some additional resources that might help:

• Online tutorials and videos
• Practice problems and worksheets
• Online calculators and tools
• Math textbooks and reference books

Conclusion

In this article, we've answered some of the most frequently asked questions about the expression (4^-1×25) and how to simplify it. We've also provided additional resources and tips to help you understand and apply the rules of exponents. Remember to always follow the steps and rules outlined above to simplify expressions with exponents.

Practice Problems

• Simplify the expression (3^-2×53).
• Simplify the expression (2^4×3-1).
• Simplify the expression (4^2×2-3).

Solutions

• (3^-2×53) = (1/9×125) = 14.44
• (2^4×3-1) = (16×1/3) = 5.33
• (4^2×2-3) = (16×1/8) = 2

Real-World Applications

• Exponents are used in many real-world applications, such as finance, science, and engineering.
• Understanding exponents is essential for solving problems in these fields.
• Simplifying expressions with exponents is a crucial skill for anyone working in these fields.

Tips and Tricks

• When dealing with negative exponents, remember to rewrite them as positive exponents by taking the reciprocal of the base.
• When combining bases, make sure to multiply the bases together.
• When multiplying numerators and denominators, make sure to multiply the numerators together and the denominators together.

Common Mistakes

• Forgetting to rewrite negative exponents as positive exponents.
• Not combining bases correctly.
• Not multiplying numerators and denominators correctly.