Simplify $5 \times 5^3$, Leaving Your Answer In Index Form.
Understanding Exponents and Multiplication
In mathematics, exponents are a shorthand way of representing repeated multiplication. When we see an expression like $5^3$, it means that 5 is multiplied by itself three times: $5 \times 5 \times 5$. This can be simplified to $125$. However, when we have an expression like $5 \times 5^3$, we need to apply the rules of exponentiation to simplify it.
The Rule of Multiplication with Exponents
When we multiply two numbers with the same base, we add their exponents. In this case, we have $5 \times 5^3$. Since the base is the same (5), we can add the exponents: $5^1 \times 5^3 = 5^{1+3} = 5^4$.
Simplifying $5^4$
Now that we have simplified the expression to $5^4$, we need to leave our answer in index form. This means that we need to write the answer as a power of 5, rather than multiplying the numbers together.
Why Leave the Answer in Index Form?
Leaving the answer in index form is useful because it allows us to easily compare and manipulate expressions with exponents. For example, if we have two expressions like $5^4$ and $5^5$, we can easily compare them by looking at their exponents. This is much easier than multiplying the numbers together and comparing the results.
Real-World Applications of Exponents
Exponents have many real-world applications, including finance, science, and engineering. For example, in finance, exponents are used to calculate compound interest. In science, exponents are used to describe the growth and decay of populations. In engineering, exponents are used to describe the behavior of complex systems.
Conclusion
In conclusion, simplifying $5 \times 5^3$ involves applying the rules of exponentiation to add the exponents and leave the answer in index form. This is a useful skill to have, as it allows us to easily compare and manipulate expressions with exponents. By understanding exponents and how to simplify expressions with them, we can solve a wide range of mathematical problems and apply our knowledge to real-world situations.
Example Problems
Here are a few example problems that involve simplifying expressions with exponents:
Practice Problems
Here are a few practice problems that involve simplifying expressions with exponents:
- Simplify $6 \times 6^2$
- Simplify $7 \times 7^3$
- Simplify $8 \times 8^4$
Answer Key
Here are the answers to the practice problems:
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8 \times 8^4 = 8^{1+4} = 8^5 = 32768$<br/>
Q: What is the rule for multiplying two numbers with the same base?
A: When we multiply two numbers with the same base, we add their exponents. For example, $5^1 \times 5^3 = 5^{1+3} = 5^4$.
Q: How do I simplify an expression like $2 \times 2^4$?
A: To simplify an expression like $2 \times 2^4$, we add the exponents: $2^1 \times 2^4 = 2^{1+4} = 2^5$.
Q: What is the difference between $5^4$ and $5^5$?
A: $5^4$ and $5^5$ are two different expressions with the same base (5). The only difference is the exponent. $5^4$ means that 5 is multiplied by itself four times, while $5^5$ means that 5 is multiplied by itself five times.
Q: Can I simplify an expression like $3 \times 3^2$?
A: Yes, we can simplify an expression like $3 \times 3^2$ by adding the exponents: $3^1 \times 3^2 = 3^{1+2} = 3^3$.
Q: How do I simplify an expression like $4 \times 4^3$?
A: To simplify an expression like $4 \times 4^3$, we add the exponents: $4^1 \times 4^3 = 4^{1+3} = 4^4$.
Q: What is the rule for dividing two numbers with the same base?
A: When we divide two numbers with the same base, we subtract their exponents. For example, $5^4 \div 5^3 = 5^{4-3} = 5^1$.
Q: How do I simplify an expression like $2^4 \div 2^3$?
A: To simplify an expression like $2^4 \div 2^3$, we subtract the exponents: $2^4 \div 2^3 = 2^{4-3} = 2^1$.
Q: Can I simplify an expression like $3^5 \div 3^2$?
A: Yes, we can simplify an expression like $3^5 \div 3^2$ by subtracting the exponents: $3^5 \div 3^2 = 3^{5-2} = 3^3$.
Q: How do I simplify an expression like $4^6 \div 4^3$?
A: To simplify an expression like $4^6 \div 4^3$, we subtract the exponents: $4^6 \div 4^3 = 4^{6-3} = 4^3$.
Conclusion
In conclusion, simplifying expressions with exponents involves applying the rules of exponentiation to add or subtract exponents. By understanding these rules, we can simplify complex expressions and solve a wide range of mathematical problems.
Practice Problems
Here are a few practice problems that involve simplifying expressions with exponents:
- Simplify $6 \times 6^2$
- Simplify $7 \times 7^3$
- Simplify $8 \times 8^4$
- Simplify $2^4 \div 2^3$
- Simplify $3^5 \div 3^2$
Answer Key
Here are the answers to the practice problems: