What Is The Value Of $k$ In The Product Of Powers Below?$10^{-3} \cdot 10 \cdot 10^k = 10^{-3} = \frac{1}{10^3}$A. -3 B. -1 C. 0 D. 1

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Introduction

In mathematics, the product of powers is a fundamental concept that helps us simplify complex expressions involving exponents. When dealing with the product of powers, we need to apply the rule that states when multiplying two powers with the same base, we add their exponents. In this article, we will explore the value of k in the product of powers given by the equation: $10^{-3} \cdot 10 \cdot 10^k = 10^{-3} = \frac{1}{10^3}$

The Product of Powers Rule

The product of powers rule states that when multiplying two powers with the same base, we add their exponents. Mathematically, this can be represented as:

am⋅an=am+na^m \cdot a^n = a^{m+n}

where a is the base and m and n are the exponents.

Applying the Product of Powers Rule

In the given equation, we have three terms: $10^{-3}$, $10$, and $10^k$. We can apply the product of powers rule to simplify the expression:

10−3⋅10⋅10k=10−3+1+k10^{-3} \cdot 10 \cdot 10^k = 10^{-3 + 1 + k}

Using the rule that $10^{-3 + 1 + k} = 10^{(-3 + 1 + k)}$

Simplifying the Expression

Now, let's simplify the expression further by combining the exponents:

10−3+1+k=10(−3+1+k)10^{-3 + 1 + k} = 10^{(-3 + 1 + k)}

10−3+1+k=10(−2+k)10^{-3 + 1 + k} = 10^{(-2 + k)}

Equating the Exponents

Since the given equation is equal to $10^{-3}$, we can equate the exponents:

−2+k=−3-2 + k = -3

Solving for k

To solve for k, we need to isolate the variable k on one side of the equation. We can do this by adding 2 to both sides of the equation:

k=−3+2k = -3 + 2

k=−1k = -1

Conclusion

In conclusion, the value of k in the product of powers given by the equation: $10^{-3} \cdot 10 \cdot 10^k = 10^{-3} = \frac{1}{10^3}$ is -1.

Answer

The correct answer is B. -1.

Discussion

This problem requires a deep understanding of the product of powers rule and how to apply it to simplify complex expressions. The key concept here is to recognize that when multiplying two powers with the same base, we add their exponents. By applying this rule, we can simplify the expression and solve for the value of k.

Related Topics

  • Product of powers rule
  • Exponents
  • Simplifying expressions
  • Algebra

References

Q: What is the product of powers rule?

A: The product of powers rule states that when multiplying two powers with the same base, we add their exponents. Mathematically, this can be represented as:

am⋅an=am+na^m \cdot a^n = a^{m+n}

where a is the base and m and n are the exponents.

Q: How do I apply the product of powers rule?

A: To apply the product of powers rule, simply add the exponents of the two powers with the same base. For example:

102⋅103=102+3=10510^2 \cdot 10^3 = 10^{2+3} = 10^5

Q: What if the bases are different?

A: If the bases are different, we cannot apply the product of powers rule. For example:

102⋅53=?10^2 \cdot 5^3 = ?

In this case, we need to multiply the numbers as they are, without adding the exponents.

Q: Can I apply the product of powers rule to negative exponents?

A: Yes, you can apply the product of powers rule to negative exponents. For example:

10−2⋅10−3=10−2−3=10−510^{-2} \cdot 10^{-3} = 10^{-2-3} = 10^{-5}

Q: How do I simplify expressions involving exponents?

A: To simplify expressions involving exponents, you can apply the product of powers rule, as well as other rules such as:

  • Power of a power rule: $(am)n = a^{m \cdot n}$
  • Product of powers rule: $a^m \cdot a^n = a^{m+n}$
  • Quotient of powers rule: $\frac{am}{an} = a^{m-n}$

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

A: Some common mistakes to avoid when working with exponents include:

  • Not applying the product of powers rule correctly: Make sure to add the exponents when multiplying powers with the same base.
  • Not simplifying expressions correctly: Make sure to apply the correct rules to simplify expressions involving exponents.
  • Not handling negative exponents correctly: Make sure to handle negative exponents correctly by applying the product of powers rule.

Q: How can I practice working with exponents?

A: You can practice working with exponents by:

  • Solving problems involving exponents: Try solving problems that involve exponents, such as simplifying expressions or solving equations.
  • Using online resources: Use online resources such as Khan Academy or Math Open Reference to practice working with exponents.
  • Working with a tutor or teacher: Work with a tutor or teacher to get help and practice working with exponents.

Conclusion

In conclusion, the product of powers rule is a fundamental concept in mathematics that helps us simplify complex expressions involving exponents. By understanding and applying this rule, we can solve problems involving exponents and simplify expressions. Remember to avoid common mistakes and practice working with exponents to become proficient.

Related Topics

  • Product of powers rule
  • Exponents
  • Simplifying expressions
  • Algebra

References