Simplify. Express Your Answer Using Positive Exponents.$\[ \frac{j^9 K}{j^0 K^{-2}} \\]

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Understanding Exponents and Simplification

In mathematics, exponents are a shorthand way of representing repeated multiplication. For example, j3j^3 means j×j×jj \times j \times j. When simplifying expressions with exponents, it's essential to understand the rules of exponentiation and how to apply them to simplify complex expressions.

The Rules of Exponentiation

There are several rules of exponentiation that we need to follow when simplifying expressions:

  • Product of Powers Rule: When multiplying two powers with the same base, we add the exponents. For example, j3×j2=j3+2=j5j^3 \times j^2 = j^{3+2} = j^5.
  • Power of a Power Rule: When raising a power to another power, we multiply the exponents. For example, (j3)2=j3×2=j6(j^3)^2 = j^{3 \times 2} = j^6.
  • Quotient of Powers Rule: When dividing two powers with the same base, we subtract the exponents. For example, j3j2=j3−2=j1\frac{j^3}{j^2} = j^{3-2} = j^1.
  • Zero Exponent Rule: Any non-zero number raised to the power of zero is equal to 1. For example, j0=1j^0 = 1.

Simplifying the Given Expression

Now that we have a good understanding of the rules of exponentiation, let's simplify the given expression:

j9kj0k−2\frac{j^9 k}{j^0 k^{-2}}

To simplify this expression, we need to apply the rules of exponentiation. We can start by simplifying the numerator and denominator separately.

Simplifying the Numerator

The numerator is j9kj^9 k. Since there are no other powers of jj or kk in the numerator, we can leave it as is.

Simplifying the Denominator

The denominator is j0k−2j^0 k^{-2}. We can simplify the denominator by applying the zero exponent rule and the quotient of powers rule.

  • Zero Exponent Rule: j0=1j^0 = 1
  • Quotient of Powers Rule: k−21=k−2\frac{k^{-2}}{1} = k^{-2}

So, the simplified denominator is k−2k^{-2}.

Combining the Numerator and Denominator

Now that we have simplified the numerator and denominator, we can combine them to get the final simplified expression.

j9kj0k−2=j9kk−2\frac{j^9 k}{j^0 k^{-2}} = \frac{j^9 k}{k^{-2}}

To simplify this expression further, we can apply the quotient of powers rule.

j9kk−2=j9k1−(−2)=j9k3\frac{j^9 k}{k^{-2}} = j^9 k^{1-(-2)} = j^9 k^3

Therefore, the simplified expression is j9k3j^9 k^3.

Conclusion

In this article, we learned how to simplify expressions with positive exponents using the rules of exponentiation. We applied the product of powers rule, power of a power rule, quotient of powers rule, and zero exponent rule to simplify the given expression. We also learned how to combine the numerator and denominator to get the final simplified expression. With practice and patience, you can become proficient in simplifying expressions with positive exponents.

Common Mistakes to Avoid

When simplifying expressions with positive exponents, there are several common mistakes to avoid:

  • Not applying the rules of exponentiation: Make sure to apply the rules of exponentiation, such as the product of powers rule, power of a power rule, quotient of powers rule, and zero exponent rule.
  • Not simplifying the numerator and denominator separately: Simplify the numerator and denominator separately before combining them.
  • Not checking for errors: Double-check your work to ensure that you have not made any errors.

Practice Problems

To practice simplifying expressions with positive exponents, try the following problems:

  • j4k2j2k−1\frac{j^4 k^2}{j^2 k^{-1}}
  • j3k4j0k2\frac{j^3 k^4}{j^0 k^2}
  • j2k3j1k−2\frac{j^2 k^3}{j^1 k^{-2}}

Answer Key

  • j4k2j2k−1=j2k3\frac{j^4 k^2}{j^2 k^{-1}} = j^2 k^3
  • j3k4j0k2=j3k2\frac{j^3 k^4}{j^0 k^2} = j^3 k^2
  • j2k3j1k−2=j1k5\frac{j^2 k^3}{j^1 k^{-2}} = j^1 k^5

Frequently Asked Questions

In this article, we will answer some frequently asked questions about simplifying expressions with positive exponents.

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

A: A positive exponent is a power that is raised to a positive number, while a negative exponent is a power that is raised to a negative number. For example, j3j^3 is a positive exponent, while j−3j^{-3} is a negative exponent.

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

A: To simplify an expression with a negative exponent, you can use the quotient of powers rule. For example, j−3k2=k2j3\frac{j^{-3}}{k^2} = \frac{k^2}{j^3}.

Q: What is the zero exponent rule?

A: The zero exponent rule states that any non-zero number raised to the power of zero is equal to 1. For example, j0=1j^0 = 1.

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

A: To simplify an expression with a zero exponent, you can use the zero exponent rule. For example, j3j0=j3\frac{j^3}{j^0} = j^3.

Q: What is the product of powers rule?

A: The product of powers rule states that when multiplying two powers with the same base, you add the exponents. For example, j3×j2=j3+2=j5j^3 \times j^2 = j^{3+2} = j^5.

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

A: To simplify an expression with multiple powers, you can use the product of powers rule. For example, j3k2j2k−1=j3−2k2−(−1)=j1k3\frac{j^3 k^2}{j^2 k^{-1}} = j^{3-2} k^{2-(-1)} = j^1 k^3.

Q: What is the power of a power rule?

A: The power of a power rule states that when raising a power to another power, you multiply the exponents. For example, (j3)2=j3×2=j6(j^3)^2 = j^{3 \times 2} = j^6.

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

A: To simplify an expression with multiple powers and a power of a power, you can use the power of a power rule. For example, (j3k2)2=j3×2k2×2=j6k4(j^3 k^2)^2 = j^{3 \times 2} k^{2 \times 2} = j^6 k^4.

Q: What is the quotient of powers rule?

A: The quotient of powers rule states that when dividing two powers with the same base, you subtract the exponents. For example, j3j2=j3−2=j1\frac{j^3}{j^2} = j^{3-2} = j^1.

Q: How do I simplify an expression with multiple powers and a quotient of powers?

A: To simplify an expression with multiple powers and a quotient of powers, you can use the quotient of powers rule. For example, j3k2j2k−1=j3−2k2−(−1)=j1k3\frac{j^3 k^2}{j^2 k^{-1}} = j^{3-2} k^{2-(-1)} = j^1 k^3.

Conclusion

In this article, we answered some frequently asked questions about simplifying expressions with positive exponents. We covered topics such as negative exponents, zero exponents, product of powers, power of a power, and quotient of powers. By understanding these rules and practicing simplifying expressions with positive exponents, you can become proficient in simplifying complex expressions and solve problems with ease.

Common Mistakes to Avoid

When simplifying expressions with positive exponents, there are several common mistakes to avoid:

  • Not applying the rules of exponentiation: Make sure to apply the rules of exponentiation, such as the product of powers rule, power of a power rule, quotient of powers rule, and zero exponent rule.
  • Not simplifying the numerator and denominator separately: Simplify the numerator and denominator separately before combining them.
  • Not checking for errors: Double-check your work to ensure that you have not made any errors.

Practice Problems

To practice simplifying expressions with positive exponents, try the following problems:

  • j4k2j2k−1\frac{j^4 k^2}{j^2 k^{-1}}
  • j3k4j0k2\frac{j^3 k^4}{j^0 k^2}
  • j2k3j1k−2\frac{j^2 k^3}{j^1 k^{-2}}

Answer Key

  • j4k2j2k−1=j2k3\frac{j^4 k^2}{j^2 k^{-1}} = j^2 k^3
  • j3k4j0k2=j3k2\frac{j^3 k^4}{j^0 k^2} = j^3 k^2
  • j2k3j1k−2=j1k5\frac{j^2 k^3}{j^1 k^{-2}} = j^1 k^5

By following the rules of exponentiation and practicing simplifying expressions with positive exponents, you can become proficient in simplifying complex expressions and solve problems with ease.