Simplify. Express The Answer Using Positive Exponents. − 2 A 2 B 4 4 A B − 8 , A ≠ 0 , B ≠ 0 -\frac{2 A^2 B^4}{4 A B^{-8}}, \ A \neq 0, \ B \neq 0 − 4 A B − 8 2 A 2 B 4 ​ , A  = 0 , B  = 0 A. − 1 2 A B − 12 -\frac{1}{2} A B^{-12} − 2 1 ​ A B − 12 B. − A 2 B − 4 -\frac{a}{2 B^{-4}} − 2 B − 4 A ​ C. − A 2 B 4 -\frac{a}{2 B^4} − 2 B 4 A ​ D. − 1 2 A B 12 -\frac{1}{2} A B^{12} − 2 1 ​ A B 12

by ADMIN 393 views

Understanding Exponents and Negative Exponents

In mathematics, exponents are a shorthand way of representing repeated multiplication. For example, a2a^2 means a×aa \times a, and a3a^3 means a×a×aa \times a \times a. When we have a negative exponent, it means we are taking the reciprocal of the expression. For instance, a−2a^{-2} means 1a2\frac{1}{a^2}.

Simplifying the Given Expression

The given expression is −2a2b44ab−8-\frac{2 a^2 b^4}{4 a b^{-8}}. To simplify this expression, we need to use the rules of exponents. We can start by canceling out any common factors in the numerator and denominator.

Step 1: Cancel Out Common Factors

We can see that both the numerator and denominator have a factor of 2a2a. We can cancel out these common factors:

−2a2b44ab−8=−a2b42ab−8-\frac{2 a^2 b^4}{4 a b^{-8}} = -\frac{a^2 b^4}{2 a b^{-8}}

Step 2: Simplify the Expression Using Exponent Rules

Now, we can use the exponent rules to simplify the expression. We know that when we divide two powers with the same base, we subtract the exponents. In this case, we have a2a^2 and aa, so we can subtract the exponents:

−a2b42ab−8=−a2−1b42b−8-\frac{a^2 b^4}{2 a b^{-8}} = -\frac{a^{2-1} b^4}{2 b^{-8}}

Step 3: Simplify the Expression Further

We can simplify the expression further by applying the exponent rule for division:

−a2−1b42b−8=−ab42b−8-\frac{a^{2-1} b^4}{2 b^{-8}} = -\frac{a b^4}{2 b^{-8}}

Step 4: Apply the Rule for Negative Exponents

Now, we can apply the rule for negative exponents, which states that a−n=1ana^{-n} = \frac{1}{a^n}. In this case, we have b−8b^{-8}, so we can rewrite it as 1b8\frac{1}{b^8}:

−ab42b−8=−ab42⋅1b8-\frac{a b^4}{2 b^{-8}} = -\frac{a b^4}{2 \cdot \frac{1}{b^8}}

Step 5: Simplify the Expression Using the Rule for Division by a Fraction

We can simplify the expression further by applying the rule for division by a fraction, which states that a1b=ab\frac{a}{\frac{1}{b}} = ab. In this case, we have −ab42⋅1b8-\frac{a b^4}{2 \cdot \frac{1}{b^8}}, so we can rewrite it as −ab42⋅b8-\frac{a b^4}{2} \cdot b^8:

−ab42⋅1b8=−ab42⋅b8-\frac{a b^4}{2 \cdot \frac{1}{b^8}} = -\frac{a b^4}{2} \cdot b^8

Step 6: Simplify the Expression Using the Rule for Multiplication

We can simplify the expression further by applying the rule for multiplication, which states that am⋅an=am+na^m \cdot a^n = a^{m+n}. In this case, we have −ab42⋅b8-\frac{a b^4}{2} \cdot b^8, so we can rewrite it as −ab4+82-\frac{a b^{4+8}}{2}:

−ab42⋅b8=−ab4+82-\frac{a b^4}{2} \cdot b^8 = -\frac{a b^{4+8}}{2}

Step 7: Simplify the Expression Using the Rule for Exponents

We can simplify the expression further by applying the rule for exponents, which states that am⋅an=am+na^m \cdot a^n = a^{m+n}. In this case, we have −ab4+82-\frac{a b^{4+8}}{2}, so we can rewrite it as −ab122-\frac{a b^{12}}{2}:

−ab4+82=−ab122-\frac{a b^{4+8}}{2} = -\frac{a b^{12}}{2}

Step 8: Write the Final Answer

The final answer is −ab122-\frac{a b^{12}}{2}.

Conclusion

In this article, we simplified the expression −2a2b44ab−8-\frac{2 a^2 b^4}{4 a b^{-8}} using the rules of exponents. We canceled out common factors, applied the exponent rules, and simplified the expression further using the rule for negative exponents and the rule for division by a fraction. The final answer is −ab122-\frac{a b^{12}}{2}.

Answer Key

Understanding Exponents and Negative Exponents

In mathematics, exponents are a shorthand way of representing repeated multiplication. For example, a2a^2 means a×aa \times a, and a3a^3 means a×a×aa \times a \times a. When we have a negative exponent, it means we are taking the reciprocal of the expression. For instance, a−2a^{-2} means 1a2\frac{1}{a^2}.

Q&A: Simplifying Expressions Using Positive Exponents

Q: What is the rule for simplifying expressions with negative exponents? A: The rule for simplifying expressions with negative exponents is to rewrite the negative exponent as a positive exponent by taking the reciprocal of the expression. For example, a−2a^{-2} means 1a2\frac{1}{a^2}.

Q: How do I simplify an expression with a negative exponent in the denominator? A: To simplify an expression with a negative exponent in the denominator, you can rewrite the negative exponent as a positive exponent by taking the reciprocal of the expression. For example, 1a−2\frac{1}{a^{-2}} means a2a^2.

Q: What is the rule for multiplying and dividing expressions with exponents? A: The rule for multiplying and dividing expressions with exponents is to add the exponents when multiplying and subtract the exponents when dividing. For example, a2⋅a3=a2+3=a5a^2 \cdot a^3 = a^{2+3} = a^5 and a2a3=a2−3=a−1\frac{a^2}{a^3} = a^{2-3} = a^{-1}.

Q: How do I simplify an expression with a negative exponent in the numerator? A: To simplify an expression with a negative exponent in the numerator, you can rewrite the negative exponent as a positive exponent by taking the reciprocal of the expression. For example, a−2a^{-2} means 1a2\frac{1}{a^2}.

Q: What is the rule for simplifying expressions with multiple exponents? A: The rule for simplifying expressions with multiple exponents is to apply the exponent rules in the correct order. For example, a2⋅a3⋅a4=a2+3+4=a9a^2 \cdot a^3 \cdot a^4 = a^{2+3+4} = a^9.

Q: How do I simplify an expression with a negative exponent in the denominator and a positive exponent in the numerator? A: To simplify an expression with a negative exponent in the denominator and a positive exponent in the numerator, you can rewrite the negative exponent as a positive exponent by taking the reciprocal of the expression. For example, a2a−3=a2−(−3)=a5\frac{a^2}{a^{-3}} = a^{2-(-3)} = a^5.

Q: What is the rule for simplifying expressions with fractional exponents? A: The rule for simplifying expressions with fractional exponents is to rewrite the fractional exponent as a product of a power and a root. For example, a12=aa^{\frac{1}{2}} = \sqrt{a} and a13=a3a^{\frac{1}{3}} = \sqrt[3]{a}.

Conclusion

In this article, we answered some common questions about simplifying expressions using positive exponents. We covered the rules for simplifying expressions with negative exponents, multiplying and dividing expressions with exponents, and simplifying expressions with multiple exponents. We also covered the rules for simplifying expressions with fractional exponents. By following these rules, you can simplify expressions with exponents and make them easier to work with.

Answer Key

  • Q1: The rule for simplifying expressions with negative exponents is to rewrite the negative exponent as a positive exponent by taking the reciprocal of the expression.
  • Q2: To simplify an expression with a negative exponent in the denominator, you can rewrite the negative exponent as a positive exponent by taking the reciprocal of the expression.
  • Q3: The rule for multiplying and dividing expressions with exponents is to add the exponents when multiplying and subtract the exponents when dividing.
  • Q4: To simplify an expression with a negative exponent in the numerator, you can rewrite the negative exponent as a positive exponent by taking the reciprocal of the expression.
  • Q5: The rule for simplifying expressions with multiple exponents is to apply the exponent rules in the correct order.
  • Q6: To simplify an expression with a negative exponent in the denominator and a positive exponent in the numerator, you can rewrite the negative exponent as a positive exponent by taking the reciprocal of the expression.
  • Q7: The rule for simplifying expressions with fractional exponents is to rewrite the fractional exponent as a product of a power and a root.