{ \left(2 B {-5}\right) 3$}$Which Of The Following Is Equivalent To The Given Expression For All { B \neq 0$}$?Choose 1 Answer:A. { \frac{2}{b^{15}}$}$B. { \frac{8}{b^{15}}$} C . \[ C. \[ C . \[ \frac{1}{2

Introduction

Exponential expressions are a fundamental concept in mathematics, and simplifying them is a crucial skill for students and professionals alike. In this article, we will explore the process of simplifying exponential expressions, with a focus on the given expression $\left(2 b^{-5}\right)^3$. We will examine the properties of exponents and apply them to simplify the expression, ultimately determining which of the given options is equivalent to the original expression.

Understanding Exponents

Before we dive into the simplification process, it's essential to understand the properties of exponents. An exponent is a small number that is placed above and to the right of a base number, indicating how many times the base number should be multiplied by itself. For example, in the expression $2^3$, the exponent $3$ indicates that the base number $2$ should be multiplied by itself three times: $2 \times 2 \times 2 = 8$.

The Power of a Power Property

One of the key properties of exponents is the power of a power property, which states that when a power is raised to another power, the exponents are multiplied. This property can be expressed mathematically as:

$(a^m)^n = a^{m \cdot n}$

where $a$ is the base number, $m$ is the first exponent, and $n$ is the second exponent.

Simplifying the Given Expression

Now that we have a solid understanding of exponents and the power of a power property, we can apply these concepts to simplify the given expression $\left(2 b^{-5}\right)^3$. Using the power of a power property, we can rewrite the expression as:

$\left(2 b^{-5}\right)^3 = 2^3 \cdot \left(b^{-5}\right)^3$

Applying the Power of a Power Property

Next, we can apply the power of a power property to simplify the expression further. Since the exponent $3$ is being raised to the power of $-5$, we can multiply the exponents:

$2^3 \cdot \left(b^{-5}\right)^3 = 2^3 \cdot b^{-5 \cdot 3}$

Simplifying the Exponents

Now that we have multiplied the exponents, we can simplify the expression further. The exponent $-5 \cdot 3$ can be evaluated as:

$-5 \cdot 3 = -15$

So, the expression becomes:

$2^3 \cdot b^{-15}$

Rewriting the Expression

Finally, we can rewrite the expression in a more simplified form. Since $2^3 = 8$, we can rewrite the expression as:

$\frac{8}{b^{15}}$

Conclusion

In conclusion, we have successfully simplified the given expression $\left(2 b^{-5}\right)^3$ using the power of a power property and the properties of exponents. The simplified expression is $\frac{8}{b^{15}}$, which is equivalent to the original expression for all $b \neq 0$. This result is consistent with option B, which is the correct answer.

Answer

The correct answer is:

B. $\frac{8}{b^{15}}$

Additional Examples

To further reinforce our understanding of simplifying exponential expressions, let's consider a few additional examples:

• $\left(3 x^2\right)^4 = 3^4 \cdot \left(x^2\right)^4 = 81x^8$
• $\left(2 y^{-3}\right)^2 = 2^2 \cdot \left(y^{-3}\right)^2 = 4y^{-6} = \frac{4}{y^6}$
• $\left(5 z^3\right)^5 = 5^5 \cdot \left(z^3\right)^5 = 3125z^{15}$

These examples demonstrate the application of the power of a power property and the properties of exponents to simplify complex exponential expressions.

Final Thoughts

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Q: What is the power of a power property?

A: The power of a power property is a fundamental concept in mathematics that states that when a power is raised to another power, the exponents are multiplied. This property can be expressed mathematically as:

$(a^m)^n = a^{m \cdot n}$

where $a$ is the base number, $m$ is the first exponent, and $n$ is the second exponent.

Q: How do I apply the power of a power property to simplify an expression?

A: To apply the power of a power property, simply multiply the exponents of the base number. For example, if you have the expression $\left(2 b^{-5}\right)^3$, you can rewrite it as:

$\left(2 b^{-5}\right)^3 = 2^3 \cdot \left(b^{-5}\right)^3$

Then, multiply the exponents:

$2^3 \cdot \left(b^{-5}\right)^3 = 2^3 \cdot b^{-5 \cdot 3}$

Finally, simplify the expression:

$2^3 \cdot b^{-5 \cdot 3} = 8 \cdot b^{-15} = \frac{8}{b^{15}}$

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

A: A positive exponent indicates that the base number should be multiplied by itself a certain number of times. For example, in the expression $2^3$, the exponent $3$ indicates that the base number $2$ should be multiplied by itself three times: $2 \times 2 \times 2 = 8$.

A negative exponent, on the other hand, indicates that the base number should be divided by itself a certain number of times. For example, in the expression $2^{-3}$, the exponent $-3$ indicates that the base number $2$ should be divided by itself three times: $\frac{1}{2} \div \frac{1}{2} \div \frac{1}{2} = \frac{1}{8}$.

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

A: To simplify an expression with a negative exponent, simply rewrite the expression with a positive exponent. For example, if you have the expression $2^{-3}$, you can rewrite it as:

$2^{-3} = \frac{1}{2^3}$

Then, simplify the expression:

$\frac{1}{2^3} = \frac{1}{8}$

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. This rule can be expressed mathematically as:

$a^0 = 1$

where $a$ is any non-zero number.

Q: How do I apply the zero-exponent rule to simplify an expression?

A: To apply the zero-exponent rule, simply rewrite the expression with a zero exponent. For example, if you have the expression $2^0$, you can rewrite it as:

$2^0 = 1$

Q: What is the product of powers property?

A: The product of powers property states that when multiplying two powers with the same base, the exponents are added. This property can be expressed mathematically as:

$a^m \cdot a^n = a^{m + n}$

where $a$ is the base number, $m$ is the first exponent, and $n$ is the second exponent.

Q: How do I apply the product of powers property to simplify an expression?

A: To apply the product of powers property, simply add the exponents of the base number. For example, if you have the expression $2^3 \cdot 2^4$, you can rewrite it as:

$2^3 \cdot 2^4 = 2^{3 + 4}$

Then, simplify the expression:

$2^{3 + 4} = 2^7 = 128$

Conclusion

In conclusion, simplifying exponential expressions is a crucial skill for students and professionals alike. By understanding the properties of exponents and applying the power of a power property, the product of powers property, and the zero-exponent rule, we can simplify complex expressions and arrive at the correct solution. We hope that this article has provided valuable insights and examples to help you master the art of simplifying exponential expressions.