Select The Correct Answer.Which Expression Is Equivalent To The Given Expression?$\[ \frac{6 A B}{(a^0 B^2)^4} \\]A. \[$\frac{6}{a^3 B^6}\$\]B. \[$\frac{6}{a^5 B^7}\$\]C. \[$\frac{6 A}{b^5}\$\]D. \[$\frac{6

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Understanding Exponents and Their Properties

In algebra, exponents are a crucial concept that helps us simplify complex expressions and solve equations. An exponent is a small number that is placed above and to the right of a number or a variable, indicating how many times the base is multiplied by itself. For example, in the expression a3a^3, the exponent 3 indicates that the base aa is multiplied by itself three times, resulting in a×a×aa \times a \times a.

Simplifying the Given Expression

The given expression is 6ab(a0b2)4\frac{6 a b}{(a^0 b^2)^4}. To simplify this expression, we need to apply the properties of exponents. Let's start by simplifying the denominator using the property (am)n=am×n(a^m)^n = a^{m \times n}.

6ab(a0b2)4=6aba0×4b2×4\frac{6 a b}{(a^0 b^2)^4} = \frac{6 a b}{a^{0 \times 4} b^{2 \times 4}}

Applying the Zero Exponent Rule

In the expression a0×4a^{0 \times 4}, the exponent 0 multiplied by 4 results in 0. According to the zero exponent rule, any non-zero number raised to the power of 0 is equal to 1. Therefore, a0×4=a0=1a^{0 \times 4} = a^0 = 1.

6aba0×4b2×4=6ab1×b8\frac{6 a b}{a^{0 \times 4} b^{2 \times 4}} = \frac{6 a b}{1 \times b^8}

Simplifying the Expression Further

Now, we can simplify the expression by canceling out the common factors in the numerator and the denominator.

6ab1×b8=6ab7\frac{6 a b}{1 \times b^8} = \frac{6 a}{b^7}

Comparing the Simplified Expression with the Options

Now that we have simplified the expression, let's compare it with the options provided.

A. 6a3b6\frac{6}{a^3 b^6} B. 6a5b7\frac{6}{a^5 b^7} C. 6ab5\frac{6 a}{b^5} D. 6ab7\frac{6 a}{b^7}

The simplified expression 6ab7\frac{6 a}{b^7} matches option D.

Conclusion

In this article, we simplified the given expression 6ab(a0b2)4\frac{6 a b}{(a^0 b^2)^4} using the properties of exponents. We applied the zero exponent rule and simplified the expression further by canceling out the common factors. The final simplified expression 6ab7\frac{6 a}{b^7} matches option D.

Key Takeaways

  • Exponents are a crucial concept in algebra that helps us simplify complex expressions and solve equations.
  • The zero exponent rule states that any non-zero number raised to the power of 0 is equal to 1.
  • When simplifying expressions, we can apply the properties of exponents to cancel out common factors.

Frequently Asked Questions

  • What is the zero exponent rule? The zero exponent rule states that any non-zero number raised to the power of 0 is equal to 1.
  • How do we simplify expressions using exponents? We can simplify expressions by applying the properties of exponents, such as the zero exponent rule, and canceling out common factors.
  • What is the final simplified expression for the given expression 6ab(a0b2)4\frac{6 a b}{(a^0 b^2)^4}? The final simplified expression is 6ab7\frac{6 a}{b^7}.
    Simplifying Exponents in Algebraic Expressions: Q&A =====================================================

Frequently Asked Questions

In the previous article, we simplified the expression 6ab(a0b2)4\frac{6 a b}{(a^0 b^2)^4} using the properties of exponents. In this article, we will answer some frequently asked questions related to simplifying exponents in algebraic expressions.

Q: What is the zero exponent rule?

A: The zero exponent rule states that any non-zero number raised to the power of 0 is equal to 1. For example, a0=1a^0 = 1 for any non-zero value of aa.

Q: How do we simplify expressions using exponents?

A: We can simplify expressions by applying the properties of exponents, such as the zero exponent rule, and canceling out common factors. For example, in the expression a3b2a2b3\frac{a^3 b^2}{a^2 b^3}, we can simplify it by canceling out the common factors: a3b2a2b3=a3−2b2−31=ab−11=ab\frac{a^3 b^2}{a^2 b^3} = \frac{a^{3-2} b^{2-3}}{1} = \frac{a b^{-1}}{1} = \frac{a}{b}.

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

A: A positive exponent indicates that the base is multiplied by itself a certain number of times. For example, a3a^3 means a×a×aa \times a \times a. A negative exponent indicates that the base is divided by itself a certain number of times. For example, a−3a^{-3} means 1a×a×a\frac{1}{a \times a \times a}.

Q: How do we handle negative exponents in algebraic expressions?

A: We can handle negative exponents by rewriting them as positive exponents. For example, a−3a^{-3} can be rewritten as 1a3\frac{1}{a^3}. We can also move the negative exponent to the other side of the fraction bar. For example, a−3b2=1a3b2\frac{a^{-3}}{b^2} = \frac{1}{a^3 b^2}.

Q: What is the rule for multiplying exponents with the same base?

A: When multiplying exponents with the same base, we add the exponents. For example, a3×a2=a3+2=a5a^3 \times a^2 = a^{3+2} = a^5.

Q: What is the rule for dividing exponents with the same base?

A: When dividing exponents with the same base, we subtract the exponents. For example, a3a2=a3−2=a1=a\frac{a^3}{a^2} = a^{3-2} = a^1 = a.

Q: How do we simplify expressions with multiple exponents?

A: We can simplify expressions with multiple exponents by applying the rules for multiplying and dividing exponents. For example, in the expression a3b2a2b3\frac{a^3 b^2}{a^2 b^3}, we can simplify it by canceling out the common factors: a3b2a2b3=a3−2b2−31=ab−11=ab\frac{a^3 b^2}{a^2 b^3} = \frac{a^{3-2} b^{2-3}}{1} = \frac{a b^{-1}}{1} = \frac{a}{b}.

Conclusion

In this article, we answered some frequently asked questions related to simplifying exponents in algebraic expressions. We discussed the zero exponent rule, the difference between positive and negative exponents, and the rules for multiplying and dividing exponents with the same base. We also provided examples of how to simplify expressions with multiple exponents.

Key Takeaways

  • The zero exponent rule states that any non-zero number raised to the power of 0 is equal to 1.
  • A positive exponent indicates that the base is multiplied by itself a certain number of times.
  • A negative exponent indicates that the base is divided by itself a certain number of times.
  • When multiplying exponents with the same base, we add the exponents.
  • When dividing exponents with the same base, we subtract the exponents.

Frequently Asked Questions (FAQs)

  • What is the zero exponent rule? The zero exponent rule states that any non-zero number raised to the power of 0 is equal to 1.
  • How do we simplify expressions using exponents? We can simplify expressions by applying the properties of exponents, such as the zero exponent rule, and canceling out common factors.
  • What is the difference between a positive exponent and a negative exponent? A positive exponent indicates that the base is multiplied by itself a certain number of times, while a negative exponent indicates that the base is divided by itself a certain number of times.
  • How do we handle negative exponents in algebraic expressions? We can handle negative exponents by rewriting them as positive exponents or moving the negative exponent to the other side of the fraction bar.
  • What is the rule for multiplying exponents with the same base? When multiplying exponents with the same base, we add the exponents.
  • What is the rule for dividing exponents with the same base? When dividing exponents with the same base, we subtract the exponents.