Forgetting to add exponents

Oct 13, 2025 by magicalceylontours.com 129 views

Understanding Exponents

Exponents are a shorthand way of representing repeated multiplication of a number. In the given expression, $5^5 \times 5^2 \times 5$ , we have three instances of the number 5 being multiplied together. The exponent 5 indicates that the number 5 is being multiplied by itself 5 times. Similarly, the exponent 2 indicates that the number 5 is being multiplied by itself 2 times. The last instance of 5 is simply a single number.

Applying the Rules of Exponents

To simplify the expression, we need to apply the rules of exponents. One of the key rules is that when we multiply two numbers with the same base (in this case, 5), we add their exponents. This is known as the product rule of exponents.

Using this rule, we can rewrite the expression as:

$5^5 \times 5^2 \times 5 = 5^{5+2+1}$

Simplifying the Exponent

Now, we need to simplify the exponent by adding the numbers together.

$5^{5+2+1} = 5^8$

Conclusion

Therefore, the simplified expression using positive exponents is $5^8$ . This is the correct answer.

Why is this Important?

Understanding exponents and how to simplify expressions using positive exponents is an important concept in mathematics. It helps us to evaluate expressions more efficiently and to solve problems involving repeated multiplication.

Real-World Applications

Exponents have many real-world applications, such as:

Finance: Exponents are used to calculate compound interest and investment returns.
Science: Exponents are used to describe the growth and decay of populations, chemical reactions, and physical phenomena.
Computer Science: Exponents are used in algorithms and data structures to optimize performance and memory usage.

Common Mistakes

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

Forgetting to add exponents: When multiplying numbers with the same base, make sure to add their exponents.
Not simplifying the exponent: Make sure to simplify the exponent by adding the numbers together.
Using the wrong rule: Make sure to use the correct rule for simplifying exponents, such as the product rule or the power rule.

Practice Problems

Here are some practice problems to help you reinforce your understanding of exponents and how to simplify expressions using positive exponents:

$2^3 \times 2^4 \times 2$
$3^2 \times 3^5 \times 3$
$4^1 \times 4^2 \times 4$

Solutions

Here are the solutions to the practice problems:

$2^3 \times 2^4 \times 2 = 2^{3+4+1} = 2^8$
$3^2 \times 3^5 \times 3 = 3^{2+5+1} = 3^8$
$4^1 \times 4^2 \times 4 = 4^{1+2+1} = 4^4$

Conclusion

In conclusion, simplifying expressions using positive exponents is an important concept in mathematics. By understanding the rules of exponents and how to simplify expressions, we can evaluate expressions more efficiently and solve problems involving repeated multiplication. With practice and patience, you can become proficient in simplifying expressions using positive exponents.

Final Answer

The final answer is: B. $5^8$

Frequently Asked Questions

Q: What is the rule for simplifying exponents when multiplying numbers with the same base?

A: When multiplying numbers with the same base, we add their exponents. This is known as the product rule of exponents.

Q: How do I simplify an expression with multiple exponents?

A: To simplify an expression with multiple exponents, we need to apply the product rule of exponents. We add the exponents together and keep the same base.

Q: What is the difference between a positive exponent and a negative exponent?

A: A positive exponent indicates that the base is being multiplied by itself a certain number of times. A negative exponent indicates that the base is being divided by itself a certain number of times.

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

A: To simplify an expression with a negative exponent, we need to apply the rule that states $a^{-n} = \frac{1}{a^n}$ . This means that we can rewrite the negative exponent as a fraction with the base in the denominator.

Q: What is the rule for simplifying exponents when raising a power to a power?

A: When raising a power to a power, we multiply the exponents together. This is known as the power rule of exponents.

Q: How do I simplify an expression with multiple exponents and a power?

A: To simplify an expression with multiple exponents and a power, we need to apply the product rule and the power rule of exponents. We add the exponents together and multiply the exponents of the power.

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

A: An exponent is a small number that is written to the right of a base and indicates that the base is being multiplied by itself a certain number of times. A power is an expression that has an exponent.

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

A: To simplify an expression with a power and a negative exponent, we need to apply the rule that states $a^{-n} = \frac{1}{a^n}$ . We can rewrite the negative exponent as a fraction with the base in the denominator and then apply the power rule of exponents.

Q: What is the rule for simplifying exponents when dividing numbers with the same base?

A: When dividing numbers with the same base, we subtract their exponents. This is known as the quotient rule of exponents.

Q: How do I simplify an expression with multiple exponents and a division?

A: To simplify an expression with multiple exponents and a division, we need to apply the quotient rule of exponents. We subtract the exponents together and keep the same base.

Common Mistakes

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

Forgetting to add exponents: When multiplying numbers with the same base, make sure to add their exponents.
Not simplifying the exponent: Make sure to simplify the exponent by adding the numbers together.
Using the wrong rule: Make sure to use the correct rule for simplifying exponents, such as the product rule or the power rule.
Not considering negative exponents: Make sure to consider negative exponents and apply the rule that states $a^{-n} = \frac{1}{a^n}$ .

Practice Problems

Here are some practice problems to help you reinforce your understanding of exponents and how to simplify expressions using positive exponents:

$2^3 \times 2^4 \times 2$
$3^2 \times 3^5 \times 3$
$4^1 \times 4^2 \times 4$
$5^{-2}$
$6^3 \div 6^2$

Solutions

Here are the solutions to the practice problems:

$2^3 \times 2^4 \times 2 = 2^{3+4+1} = 2^8$
$3^2 \times 3^5 \times 3 = 3^{2+5+1} = 3^8$
$4^1 \times 4^2 \times 4 = 4^{1+2+1} = 4^4$
$5^{-2} = \frac{1}{5^2} = \frac{1}{25}$
$6^3 \div 6^2 = 6^{3-2} = 6^1 = 6$

Conclusion

Final Answer

The final answer is: B. $5^8$