Which Expression Shows The Simplified Form Of $\left(8 R^{-5}\right)^{-3}$?A. $8 R^{15}$B. $\frac{8}{r^{15}}$C. $512 R^{15}$D. $\frac{r^{15}}{512}$

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Introduction

When dealing with exponential expressions, it's essential to understand the rules of exponents to simplify them correctly. In this article, we'll focus on simplifying the expression (8rβˆ’5)βˆ’3\left(8 r^{-5}\right)^{-3} and explore the different options available.

Understanding Exponents

Exponents are a shorthand way of representing repeated multiplication. For example, a3a^3 can be written as aβ‹…aβ‹…aa \cdot a \cdot a. When dealing with negative exponents, we can rewrite them as positive exponents by taking the reciprocal of the base. For instance, aβˆ’3a^{-3} can be written as 1a3\frac{1}{a^3}.

Simplifying the Expression

To simplify the expression (8rβˆ’5)βˆ’3\left(8 r^{-5}\right)^{-3}, we need to apply the power rule of exponents, which states that (am)n=amβ‹…n(a^m)^n = a^{m \cdot n}. In this case, we have:

(8rβˆ’5)βˆ’3=8βˆ’3β‹…(rβˆ’5)βˆ’3\left(8 r^{-5}\right)^{-3} = 8^{-3} \cdot (r^{-5})^{-3}

Applying the Power Rule

Now, let's apply the power rule to each part of the expression:

8βˆ’3=183=15128^{-3} = \frac{1}{8^3} = \frac{1}{512}

(rβˆ’5)βˆ’3=r5β‹…3=r15(r^{-5})^{-3} = r^{5 \cdot 3} = r^{15}

Combining the Results

Now that we have simplified each part of the expression, we can combine them to get the final result:

(8rβˆ’5)βˆ’3=1512β‹…r15=r15512\left(8 r^{-5}\right)^{-3} = \frac{1}{512} \cdot r^{15} = \frac{r^{15}}{512}

Conclusion

In conclusion, the simplified form of (8rβˆ’5)βˆ’3\left(8 r^{-5}\right)^{-3} is r15512\frac{r^{15}}{512}. This result can be obtained by applying the power rule of exponents and simplifying each part of the expression.

Answer Options

Now that we have simplified the expression, let's compare our result with the answer options:

  • A. 8r158 r^{15}: This option is incorrect because the base is not simplified correctly.
  • B. 8r15\frac{8}{r^{15}}: This option is incorrect because the base is not simplified correctly, and the exponent is not negative.
  • C. 512r15512 r^{15}: This option is incorrect because the base is not simplified correctly, and the exponent is not negative.
  • D. r15512\frac{r^{15}}{512}: This option is correct because it matches our simplified result.

Final Answer

The final answer is r15512\boxed{\frac{r^{15}}{512}}.

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Q: What is the rule for simplifying exponential expressions?

A: The rule for simplifying exponential expressions is to apply the power rule of exponents, which states that (am)n=amβ‹…n(a^m)^n = a^{m \cdot n}.

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

A: To simplify an expression with a negative exponent, you can rewrite it as a positive exponent by taking the reciprocal of the base. For example, aβˆ’3a^{-3} can be written as 1a3\frac{1}{a^3}.

Q: What is the power rule of exponents?

A: The power rule of exponents states that (am)n=amβ‹…n(a^m)^n = a^{m \cdot n}. This means that when you raise a power to another power, you multiply the exponents.

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

A: To simplify an expression with multiple exponents, you can apply the power rule of exponents by multiplying the exponents. For example, (am)n=amβ‹…n(a^m)^n = a^{m \cdot n}.

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

A: A positive exponent indicates that the base is raised to a power, while a negative exponent indicates that the base is taken to a power. For example, a3a^3 indicates that the base aa is raised to the power of 3, while aβˆ’3a^{-3} indicates that the base aa is taken to the power of -3.

Q: How do I simplify an expression with a fraction as the base?

A: To simplify an expression with a fraction as the base, you can apply the power rule of exponents by multiplying the exponents. For example, (ab)n=anbn(\frac{a}{b})^n = \frac{a^n}{b^n}.

Q: What is the final answer to the expression (8rβˆ’5)βˆ’3\left(8 r^{-5}\right)^{-3}?

A: The final answer to the expression (8rβˆ’5)βˆ’3\left(8 r^{-5}\right)^{-3} is r15512\frac{r^{15}}{512}.

Q: Why is it important to simplify exponential expressions?

A: Simplifying exponential expressions is important because it helps to make the expression easier to understand and work with. It also helps to avoid errors and make calculations more efficient.

Q: Can you provide more examples of simplifying exponential expressions?

A: Yes, here are a few more examples:

  • (2x3)2=4x6\left(2 x^3\right)^2 = 4 x^6
  • (34yβˆ’2)3=2764yβˆ’6\left(\frac{3}{4} y^{-2}\right)^3 = \frac{27}{64} y^{-6}
  • (5z4)βˆ’2=125zβˆ’8\left(5 z^4\right)^{-2} = \frac{1}{25} z^{-8}

Q: How do I know which answer option is correct?

A: To determine which answer option is correct, you can apply the power rule of exponents and simplify the expression. If the result matches one of the answer options, then that option is correct.

Q: Can you provide a summary of the key concepts?

A: Yes, here is a summary of the key concepts:

  • The power rule of exponents states that (am)n=amβ‹…n(a^m)^n = a^{m \cdot n}.
  • A negative exponent indicates that the base is taken to a power.
  • To simplify an expression with multiple exponents, you can apply the power rule of exponents by multiplying the exponents.
  • To simplify an expression with a fraction as the base, you can apply the power rule of exponents by multiplying the exponents.

Q: What is the final answer to the expression (8rβˆ’5)βˆ’3\left(8 r^{-5}\right)^{-3} in terms of the answer options?

A: The final answer to the expression (8rβˆ’5)βˆ’3\left(8 r^{-5}\right)^{-3} is answer option D, r15512\frac{r^{15}}{512}.