Evaluate The Expression: ${ 4^2 \times 4^6 = }$

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Understanding Exponential Notation

Exponential notation is a shorthand way of writing repeated multiplication. It is commonly used to express large numbers in a more compact form. In this notation, a number raised to a power is multiplied by itself that many times. For example, 424^2 means 44 multiplied by itself 22 times, which is equal to 1616. Similarly, 464^6 means 44 multiplied by itself 66 times.

Evaluating the Expression: 42×464^2 \times 4^6

To evaluate the expression 42×464^2 \times 4^6, we need to follow the order of operations (PEMDAS). The first step is to evaluate the exponents. 424^2 is equal to 1616, and 464^6 is equal to 40964096. Now, we need to multiply these two numbers together.

Multiplying Exponential Expressions

When multiplying exponential expressions with the same base, we can add the exponents. In this case, we have 424^2 and 464^6. Since the bases are the same, we can add the exponents: 2+6=82 + 6 = 8. Therefore, 42×464^2 \times 4^6 is equal to 484^8.

Evaluating 484^8

To evaluate 484^8, we need to multiply 44 by itself 88 times. This can be done using a calculator or by multiplying the numbers together manually. 484^8 is equal to 65,53665,536.

Conclusion

In conclusion, the expression 42×464^2 \times 4^6 can be evaluated by following the order of operations and using the rule for multiplying exponential expressions with the same base. By adding the exponents, we can simplify the expression and evaluate it more easily.

Real-World Applications

Exponential expressions are used in many real-world applications, such as finance, science, and engineering. For example, in finance, exponential expressions are used to calculate compound interest. In science, exponential expressions are used to model population growth and decay. In engineering, exponential expressions are used to design and optimize systems.

Tips and Tricks

Here are some tips and tricks for evaluating exponential expressions:

  • Use the order of operations: When evaluating exponential expressions, always follow the order of operations (PEMDAS).
  • Use the rule for multiplying exponential expressions: When multiplying exponential expressions with the same base, add the exponents.
  • Use a calculator: If you need to evaluate a large exponential expression, use a calculator to simplify the calculation.
  • Practice, practice, practice: The more you practice evaluating exponential expressions, the more comfortable you will become with the rules and procedures.

Common Mistakes

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

  • Not following the order of operations: Failing to follow the order of operations can lead to incorrect results.
  • Not using the rule for multiplying exponential expressions: Failing to use the rule for multiplying exponential expressions can lead to incorrect results.
  • Not using a calculator: Failing to use a calculator when necessary can lead to incorrect results.
  • Not practicing: Failing to practice evaluating exponential expressions can lead to a lack of understanding and confidence.

Conclusion

In conclusion, evaluating exponential expressions is an important skill that is used in many real-world applications. By following the order of operations and using the rule for multiplying exponential expressions, we can simplify and evaluate exponential expressions more easily. With practice and patience, anyone can become proficient in evaluating exponential expressions.

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Q: What is the order of operations?

A: The order of operations is a set of rules that tells us which operations to perform first when we have multiple operations in an expression. The acronym PEMDAS is commonly used to remember the order of operations:

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

Q: How do I evaluate exponential expressions with the same base?

A: When evaluating exponential expressions with the same base, we can add the exponents. For example, 42×464^2 \times 4^6 can be evaluated by adding the exponents: 2+6=82 + 6 = 8. Therefore, 42×464^2 \times 4^6 is equal to 484^8.

Q: What is the rule for multiplying exponential expressions with different bases?

A: When multiplying exponential expressions with different bases, we can multiply the bases and add the exponents. For example, 23×342^3 \times 3^4 can be evaluated by multiplying the bases and adding the exponents: (2×3)3+4=67(2 \times 3)^{3+4} = 6^7.

Q: How do I evaluate exponential expressions with negative exponents?

A: When evaluating exponential expressions with negative exponents, we can rewrite the expression with a positive exponent. For example, 4−24^{-2} can be rewritten as 142=116\frac{1}{4^2} = \frac{1}{16}.

Q: What is the rule for evaluating exponential expressions with fractional exponents?

A: When evaluating exponential expressions with fractional exponents, we can rewrite the expression as a product of a number raised to a power and a number raised to a power. For example, 4124^{\frac{1}{2}} can be rewritten as 4=2\sqrt{4} = 2.

Q: How do I evaluate exponential expressions with scientific notation?

A: When evaluating exponential expressions with scientific notation, we can rewrite the expression in standard form and then evaluate it. For example, 4.5×1034.5 \times 10^3 can be rewritten as 4,500,0004,500,000 and then evaluated.

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

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

  • Not following the order of operations: Failing to follow the order of operations can lead to incorrect results.
  • Not using the rule for multiplying exponential expressions: Failing to use the rule for multiplying exponential expressions can lead to incorrect results.
  • Not using a calculator: Failing to use a calculator when necessary can lead to incorrect results.
  • Not practicing: Failing to practice evaluating exponential expressions can lead to a lack of understanding and confidence.

Q: How can I practice evaluating exponential expressions?

A: There are many ways to practice evaluating exponential expressions, including:

  • Using online resources: Websites such as Khan Academy and Mathway offer interactive exercises and practice problems to help you practice evaluating exponential expressions.
  • Working with a tutor: A tutor can provide one-on-one instruction and practice problems to help you improve your skills.
  • Solving problems on your own: Try solving problems on your own and then checking your answers with a calculator or online resource.

Q: What are some real-world applications of exponential expressions?

A: Exponential expressions have many real-world applications, including:

  • Finance: Exponential expressions are used to calculate compound interest and other financial calculations.
  • Science: Exponential expressions are used to model population growth and decay, as well as other scientific phenomena.
  • Engineering: Exponential expressions are used to design and optimize systems, such as electronic circuits and mechanical systems.

Q: How can I use exponential expressions in my everyday life?

A: Exponential expressions can be used in many everyday situations, including:

  • Calculating interest rates: Exponential expressions can be used to calculate compound interest and other financial calculations.
  • Modeling population growth: Exponential expressions can be used to model population growth and decay.
  • Designing systems: Exponential expressions can be used to design and optimize systems, such as electronic circuits and mechanical systems.