Select The Correct Answer.Use The Properties Of Exponents To Rewrite This Expression: $\[ \frac{\left(2 A^4\right)^2}{a^6 A^5} \\]What Is The Value Of The Rewritten Expression When $a = -5$?A. -20 B. -250 C. -500 D. -60

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Understanding Exponent Properties

When dealing with exponential expressions, it's essential to understand the properties of exponents. These properties allow us to simplify complex expressions and make them more manageable. In this article, we'll focus on using the properties of exponents to rewrite a given expression and then evaluate it for a specific value of the variable.

The Properties of Exponents

The properties of exponents are as follows:

  • Product of Powers Property: When multiplying two powers with the same base, we add the exponents.
  • Power of a Power Property: When raising a power to another power, we multiply the exponents.
  • Quotient of Powers Property: When dividing two powers with the same base, we subtract the exponents.

Rewriting the Expression

Using the properties of exponents, we can rewrite the given expression as follows:

(2a4)2a6a5=22a8a6a5=22a8−6−51=22a31=4a3\frac{\left(2 a^4\right)^2}{a^6 a^5} = \frac{2^2 a^{8}}{a^6 a^5} = \frac{2^2 a^{8-6-5}}{1} = \frac{2^2 a^3}{1} = 4 a^3

Evaluating the Expression

Now that we have rewritten the expression, we can evaluate it for a specific value of the variable. Let's say we want to find the value of the expression when a=−5a = -5.

4a3=4(−5)3=4(−125)=−5004 a^3 = 4 (-5)^3 = 4 (-125) = -500

Conclusion

In this article, we used the properties of exponents to rewrite a given expression and then evaluated it for a specific value of the variable. We started by understanding the properties of exponents, which allowed us to simplify the expression and make it more manageable. We then applied these properties to rewrite the expression and finally evaluated it for a specific value of the variable.

Key Takeaways

  • The properties of exponents are essential for simplifying complex exponential expressions.
  • The product of powers property allows us to add exponents when multiplying two powers with the same base.
  • The power of a power property allows us to multiply exponents when raising a power to another power.
  • The quotient of powers property allows us to subtract exponents when dividing two powers with the same base.

Final Answer

The final answer is −500\boxed{-500}.

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Q: What are the properties of exponents?

A: The properties of exponents are as follows:

  • Product of Powers Property: When multiplying two powers with the same base, we add the exponents.
  • Power of a Power Property: When raising a power to another power, we multiply the exponents.
  • Quotient of Powers Property: When dividing two powers with the same base, we subtract the exponents.

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

A: To apply the product of powers property, you need to multiply two powers with the same base. For example, if you have amâ‹…ana^m \cdot a^n, you can add the exponents to get am+na^{m+n}.

Q: How do I apply the power of a power property?

A: To apply the power of a power property, you need to raise a power to another power. For example, if you have (am)n(a^m)^n, you can multiply the exponents to get amâ‹…na^{m \cdot n}.

Q: How do I apply the quotient of powers property?

A: To apply the quotient of powers property, you need to divide two powers with the same base. For example, if you have aman\frac{a^m}{a^n}, you can subtract the exponents to get am−na^{m-n}.

Q: Can I simplify an expression with negative exponents?

A: Yes, you can simplify an expression with negative exponents. To do this, you need to use the properties of exponents to rewrite the expression with positive exponents. For example, if you have 1am\frac{1}{a^m}, you can rewrite it as a−ma^{-m}.

Q: How do I evaluate an expression with variables?

A: To evaluate an expression with variables, you need to substitute the given value of the variable into the expression. For example, if you have 4a34a^3 and a=−5a = -5, you can substitute −5-5 into the expression to get 4(−5)34(-5)^3.

Q: What is the final answer to the original problem?

A: The final answer to the original problem is −500\boxed{-500}.

Q: Can I use the properties of exponents to rewrite a complex expression?

A: Yes, you can use the properties of exponents to rewrite a complex expression. To do this, you need to apply the properties of exponents in the correct order to simplify the expression.

Q: How do I know which property to apply first?

A: To determine which property to apply first, you need to look at the expression and identify the operations that need to be performed. For example, if you have an expression with multiplication and division, you need to apply the quotient of powers property before applying the product of powers property.

Q: Can I use the properties of exponents to rewrite an expression with fractions?

A: Yes, you can use the properties of exponents to rewrite an expression with fractions. To do this, you need to apply the properties of exponents to the numerator and denominator separately.

Q: How do I evaluate an expression with fractions and variables?

A: To evaluate an expression with fractions and variables, you need to substitute the given value of the variable into the expression and then simplify the fraction. For example, if you have 4a3a2\frac{4a^3}{a^2} and a=−5a = -5, you can substitute −5-5 into the expression to get 4(−5)3(−5)2\frac{4(-5)^3}{(-5)^2}.

Q: What is the final answer to the expression with fractions and variables?

A: The final answer to the expression with fractions and variables is −500\boxed{-500}.