Select All The Expressions That Are Equivalent To $8^8 \cdot 8^7$.A. $8^{56}$B. $\frac{1}{8^{56}}$C. \$\frac{1}{8^{15}}$[/tex\]D. $\frac{1}{8^{-15}}$

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Introduction

Exponential expressions are a fundamental concept in mathematics, and understanding how to simplify them is crucial for solving various mathematical problems. In this article, we will focus on simplifying the expression $8^8 \cdot 8^7$ and select the equivalent forms from the given options.

Understanding Exponential Notation

Before we dive into simplifying the expression, let's briefly review exponential notation. Exponential notation is a shorthand way of writing repeated multiplication. For example, $a^b$ represents the product of $a$ multiplied by itself $b$ times. In other words, $a^b = a \cdot a \cdot a \cdot \ldots \cdot a$ ($b$ times).

Simplifying the Expression

Now that we have a basic understanding of exponential notation, let's simplify the expression $8^8 \cdot 8^7$. To simplify this expression, we can use the rule of exponents that states $a^m \cdot a^n = a^{m+n}$. Applying this rule to the given expression, we get:

88â‹…87=88+7=8158^8 \cdot 8^7 = 8^{8+7} = 8^{15}

Selecting Equivalent Forms

Now that we have simplified the expression to $8^{15}$, let's examine the given options and select the equivalent forms.

Option A: $8^{56}$

This option is not equivalent to $8^{15}$, as the exponents are different. Therefore, we can eliminate this option.

Option B: $\frac{1}{8^{56}}$

This option is also not equivalent to $8^{15}$, as the exponents are different and the expression is a fraction. Therefore, we can eliminate this option.

Option C: $\frac{1}{8^{15}}$

This option is equivalent to $8^{15}$, as the reciprocal of $8^{15}$ is $\frac{1}{8^{15}}$. Therefore, we can select this option as one of the equivalent forms.

Option D: $\frac{1}{8^{-15}}$

This option is also equivalent to $8^{15}$, as the reciprocal of $8^{-15}$ is $8^{15}$. Therefore, we can select this option as another equivalent form.

Conclusion

In conclusion, the equivalent forms of $8^8 \cdot 8^7$ are $8^{15}$, $\frac{1}{8^{15}}$, and $\frac{1}{8^{-15}}$. These forms demonstrate the importance of understanding exponential notation and the rules of exponents in simplifying mathematical expressions.

Final Answer

The final answer is:

  • A: Incorrect
  • B: Incorrect
  • C: Correct
  • D: Correct

Additional Tips and Resources

  • To simplify exponential expressions, use the rule of exponents that states $a^m \cdot a^n = a^{m+n}$.
  • To find the reciprocal of an exponential expression, use the rule that states $\frac{1}{a^b} = a^{-b}$.
  • For more practice problems and resources, visit the following websites:

Q: What is the rule of exponents for multiplying exponential expressions?

A: The rule of exponents for multiplying exponential expressions is $a^m \cdot a^n = a^{m+n}$. This means that when you multiply two exponential expressions with the same base, you add the exponents.

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

A: To simplify an exponential expression with a negative exponent, you can use the rule that states $a^{-b} = \frac{1}{a^b}$. This means that a negative exponent is equivalent to the reciprocal of the base raised to the positive exponent.

Q: What is the difference between an exponential expression and a power expression?

A: An exponential expression is a shorthand way of writing repeated multiplication, such as $a^b = a \cdot a \cdot a \cdot \ldots \cdot a$ ($b$ times). A power expression, on the other hand, is a specific value of an exponential expression, such as $a^b = c$.

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

A: To simplify an exponential expression with a fractional exponent, you can use the rule that states $a^{m/n} = \sqrt[n]{a^m}$. This means that a fractional exponent is equivalent to the nth root of the base raised to the positive exponent.

Q: What is the order of operations for simplifying exponential expressions?

A: The order of operations for simplifying exponential expressions is:

  1. Evaluate any exponential expressions inside parentheses.
  2. Evaluate any exponential expressions with negative exponents.
  3. Evaluate any exponential expressions with fractional exponents.
  4. Simplify any remaining exponential expressions using the rules of exponents.

Q: How do I simplify an exponential expression with multiple bases?

A: To simplify an exponential expression with multiple bases, you can use the rule that states $a^m \cdot b^n = (ab)^{m+n}$. This means that when you multiply two exponential expressions with different bases, you can combine the bases and add the exponents.

Q: What are some common mistakes to avoid when simplifying exponential expressions?

A: Some common mistakes to avoid when simplifying exponential expressions include:

  • Forgetting to use the rule of exponents for multiplying exponential expressions.
  • Not simplifying negative exponents correctly.
  • Not simplifying fractional exponents correctly.
  • Not following the order of operations.

Conclusion

In conclusion, simplifying exponential expressions requires a strong understanding of the rules of exponents and the order of operations. By following the rules and avoiding common mistakes, you can simplify even the most complex exponential expressions.

Additional Tips and Resources