Which Expression Is Equivalent To $\left(4^{-6}\right)^{-1} \times 4^{-2}$?

by ADMIN 76 views

Introduction

Exponential expressions are a fundamental concept in mathematics, and simplifying them is a crucial skill for students and professionals alike. In this article, we will explore the process of simplifying exponential expressions, with a focus on the expression (4−6)−1×4−2\left(4^{-6}\right)^{-1} \times 4^{-2}. We will break down the expression into smaller parts, apply the rules of exponents, and arrive at the final simplified form.

Understanding Exponents

Before we dive into the simplification process, let's review the basics of exponents. An exponent is a small number that is written to the upper right of a base number. It indicates how many times the base number should be multiplied by itself. For example, 434^3 means 4×4×44 \times 4 \times 4, or 6464. When we have a negative exponent, it means we are taking the reciprocal of the base number. For example, 4−34^{-3} means 143\frac{1}{4^3}, or 164\frac{1}{64}.

Simplifying the Expression

Now that we have a solid understanding of exponents, let's tackle the expression (4−6)−1×4−2\left(4^{-6}\right)^{-1} \times 4^{-2}. To simplify this expression, we will follow the order of operations (PEMDAS):

  1. Evaluate the expression inside the parentheses: (4−6)−1\left(4^{-6}\right)^{-1}
  2. Apply the power of a power rule: (4−6)−1=46\left(4^{-6}\right)^{-1} = 4^{6}
  3. Multiply the two expressions: 46×4−24^{6} \times 4^{-2}
  4. Apply the product of powers rule: 46×4−2=46−24^{6} \times 4^{-2} = 4^{6-2}
  5. Simplify the exponent: 46−2=444^{6-2} = 4^{4}

Final Simplified Form

After applying the rules of exponents, we arrive at the final simplified form: 444^{4}. This means that the expression (4−6)−1×4−2\left(4^{-6}\right)^{-1} \times 4^{-2} is equivalent to 444^{4}.

Conclusion

Simplifying exponential expressions requires a solid understanding of the rules of exponents. By following the order of operations and applying the power of a power rule, product of powers rule, and other rules, we can simplify complex expressions and arrive at the final simplified form. In this article, we explored the process of simplifying the expression (4−6)−1×4−2\left(4^{-6}\right)^{-1} \times 4^{-2} and arrived at the final simplified form: 444^{4}.

Common Mistakes to Avoid

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

  • Forgetting to apply the power of a power rule: When we have a negative exponent, we need to apply the power of a power rule to simplify the expression.
  • Not following the order of operations: We need to follow the order of operations (PEMDAS) to ensure that we simplify the expression correctly.
  • Not applying the product of powers rule: When we have multiple expressions with the same base, we need to apply the product of powers rule to simplify the expression.

Real-World Applications

Simplifying exponential expressions has many real-world applications. Here are a few examples:

  • Science and Engineering: Exponential expressions are used to model population growth, chemical reactions, and other phenomena.
  • Finance: Exponential expressions are used to calculate interest rates, investment returns, and other financial metrics.
  • Computer Science: Exponential expressions are used to model algorithms, data structures, and other computational concepts.

Practice Problems

To practice simplifying exponential expressions, try the following problems:

  • (23)−2×2−4\left(2^{3}\right)^{-2} \times 2^{-4}
  • (5−2)−3×5−1\left(5^{-2}\right)^{-3} \times 5^{-1}
  • (34)−1×3−2\left(3^{4}\right)^{-1} \times 3^{-2}

Conclusion

Q: What is the order of operations for simplifying exponential expressions?

A: The order of operations for simplifying exponential expressions is:

  1. Evaluate the expression inside the parentheses: This means that we need to simplify any expressions inside parentheses before we can simplify the rest of the expression.
  2. Apply the power of a power rule: This rule states that when we have a negative exponent, we need to apply the power of a power rule to simplify the expression.
  3. Apply the product of powers rule: This rule states that when we have multiple expressions with the same base, we need to apply the product of powers rule to simplify the expression.
  4. Simplify the exponent: This means that we need to simplify any exponents that are left after applying the previous rules.

Q: What is the power of a power rule?

A: The power of a power rule states that when we have a negative exponent, we need to apply the power of a power rule to simplify the expression. This rule is written as:

(am)n=amn(a^m)^n = a^{mn}

Q: What is the product of powers rule?

A: The product of powers rule states that when we have multiple expressions with the same base, we need to apply the product of powers rule to simplify the expression. This rule is written as:

am×an=am+na^m \times a^n = a^{m+n}

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

A: To simplify an expression with a negative exponent, you need to apply the power of a power rule. This means that you need to multiply the base by the reciprocal of the exponent.

For example, if we have the expression 2−32^{-3}, we can simplify it by multiplying the base by the reciprocal of the exponent:

2−3=123=182^{-3} = \frac{1}{2^3} = \frac{1}{8}

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

A: To simplify an expression with multiple bases, you need to apply the product of powers rule. This means that you need to multiply the bases together and add the exponents.

For example, if we have the expression 23×322^3 \times 3^2, we can simplify it by multiplying the bases together and adding the exponents:

23×32=23×32=8×9=722^3 \times 3^2 = 2^3 \times 3^2 = 8 \times 9 = 72

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

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

  • Forgetting to apply the power of a power rule: When we have a negative exponent, we need to apply the power of a power rule to simplify the expression.
  • Not following the order of operations: We need to follow the order of operations (PEMDAS) to ensure that we simplify the expression correctly.
  • Not applying the product of powers rule: When we have multiple expressions with the same base, we need to apply the product of powers rule to simplify the expression.

Q: How do I know when to use the power of a power rule and when to use the product of powers rule?

A: To determine whether to use the power of a power rule or the product of powers rule, you need to look at the expression and see if it has a negative exponent or multiple bases with the same base. If it has a negative exponent, you need to use the power of a power rule. If it has multiple bases with the same base, you need to use the product of powers rule.

Q: Can you give me some examples of simplifying exponential expressions?

A: Here are some examples of simplifying exponential expressions:

  • (23)−2×2−4=26×2−4=22=4\left(2^{3}\right)^{-2} \times 2^{-4} = 2^{6} \times 2^{-4} = 2^{2} = 4
  • (5−2)−3×5−1=56×5−1=55=3125\left(5^{-2}\right)^{-3} \times 5^{-1} = 5^{6} \times 5^{-1} = 5^{5} = 3125
  • (34)−1×3−2=3−4×3−2=3−6=1729\left(3^{4}\right)^{-1} \times 3^{-2} = 3^{-4} \times 3^{-2} = 3^{-6} = \frac{1}{729}

Conclusion

Simplifying exponential expressions is a crucial skill for students and professionals alike. By following the rules of exponents and applying the power of a power rule, product of powers rule, and other rules, we can simplify complex expressions and arrive at the final simplified form. In this article, we explored some frequently asked questions about simplifying exponential expressions and provided examples of how to simplify expressions with negative exponents and multiple bases.