Describe How $\left(2^3\right)\left(2^{-4}\right$\] Can Be Simplified.A. Multiply The Bases And Add The Exponents. Then Find The Reciprocal And Change The Sign Of The Exponent.B. Keep The Same Base And Add The Exponents. Then Multiply By -1.C.

Understanding Exponential Notation

Exponential notation is a shorthand way of expressing repeated multiplication of a number. It is written in the form $a^b$, where $a$ is the base and $b$ is the exponent. For example, $2^3$ means $2$ multiplied by itself $3$ times, or $2 \times 2 \times 2 = 8$. In this article, we will explore how to simplify exponential expressions involving the same base.

Simplifying Exponential Expressions with the Same Base

When simplifying exponential expressions with the same base, we can use the following rule:

• Multiply the bases and add the exponents: If we have two exponential expressions with the same base, we can multiply the bases and add the exponents. For example, $\left(2^3\right)\left(2^{-4}\right)$ can be simplified by multiplying the bases and adding the exponents.

Step 1: Multiply the Bases

To simplify the expression $\left(2^3\right)\left(2^{-4}\right)$, we first multiply the bases:

$$\left(2^3\right)\left(2^{-4}\right) = 2^{3+(-4)}$$

Step 2: Add the Exponents

Next, we add the exponents:

$$2^{3+(-4)} = 2^{-1}$$

Step 3: Simplify the Expression

Finally, we simplify the expression by finding the reciprocal of the base and changing the sign of the exponent:

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

Alternative Method: Using the Rule of Negative Exponents

Another way to simplify the expression $\left(2^3\right)\left(2^{-4}\right)$ is to use the rule of negative exponents:

• Keep the same base and add the exponents: If we have two exponential expressions with the same base, we can keep the same base and add the exponents. For example, $\left(2^3\right)\left(2^{-4}\right)$ can be simplified by keeping the same base and adding the exponents.

Step 1: Keep the Same Base

To simplify the expression $\left(2^3\right)\left(2^{-4}\right)$, we first keep the same base:

$$\left(2^3\right)\left(2^{-4}\right) = 2^{3+(-4)}$$

Step 2: Add the Exponents

Next, we add the exponents:

$$2^{3+(-4)} = 2^{-1}$$

Step 3: Simplify the Expression

Finally, we simplify the expression by finding the reciprocal of the base and changing the sign of the exponent:

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

Conclusion

In this article, we have explored how to simplify exponential expressions involving the same base. We have used two different methods to simplify the expression $\left(2^3\right)\left(2^{-4}\right)$, and we have found that both methods lead to the same result: $\frac{1}{2}$. By following these steps, you can simplify exponential expressions with the same base and gain a deeper understanding of exponential notation.

Frequently Asked Questions

Q: What is the rule for simplifying exponential expressions with the same base?

A: The rule is to multiply the bases and 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 of negative exponents: keep the same base and add the exponents.

Q: What is the difference between multiplying the bases and adding the exponents, and keeping the same base and adding the exponents?

A: Multiplying the bases and adding the exponents is a more general rule that applies to all exponential expressions, while keeping the same base and adding the exponents is a special case that applies only to exponential expressions with the same base.

Glossary of Terms

• Exponential notation: A shorthand way of expressing repeated multiplication of a number.
• Base: The number being multiplied by itself.
• Exponent: The number of times the base is multiplied by itself.
• Negative exponent: An exponent that is less than zero.
• Rule of negative exponents: A rule that states that an exponential expression with a negative exponent can be simplified by keeping the same base and adding the exponents.
  Exponential Expressions Q&A =============================

Frequently Asked Questions

Q: What is the rule for simplifying exponential expressions with the same base?

A: The rule is to multiply the bases and 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 of negative exponents: keep the same base and add the exponents.

Q: What is the difference between multiplying the bases and adding the exponents, and keeping the same base and adding the exponents?

A: Multiplying the bases and adding the exponents is a more general rule that applies to all exponential expressions, while keeping the same base and adding the exponents is a special case that applies only to exponential expressions with the same base.

Q: Can I simplify an exponential expression with a variable base?

A: Yes, you can simplify an exponential expression with a variable base by using the rule of exponents: multiply the bases and add the exponents.

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

A: To simplify an exponential expression with a variable exponent, you can use the rule of exponents: multiply the bases and add the exponents.

Q: What is the rule for simplifying exponential expressions with different bases?

A: The rule is to multiply the bases and add the exponents.

Q: Can I simplify an exponential expression with a fraction as the base?

A: Yes, you can simplify an exponential expression with a fraction as the base by using the rule of exponents: multiply the bases and add the exponents.

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

A: To simplify an exponential expression with a negative base, you can use the rule of exponents: multiply the bases and add the exponents.

Q: What is the rule for simplifying exponential expressions with a zero exponent?

A: The rule is that any number raised to the power of zero is equal to 1.

Q: Can I simplify an exponential expression with a negative exponent and a variable base?

A: Yes, you can simplify an exponential expression with a negative exponent and a variable base by using the rule of negative exponents: keep the same base and add the exponents.

Q: How do I simplify an exponential expression with a variable exponent and a variable base?

A: To simplify an exponential expression with a variable exponent and a variable base, you can use the rule of exponents: multiply the bases and add the exponents.

Common Mistakes to Avoid

• Not following the order of operations: When simplifying exponential expressions, it's essential to follow the order of operations (PEMDAS): parentheses, exponents, multiplication and division, and addition and subtraction.
• Not using the correct rule for simplifying exponential expressions: Make sure to use the correct rule for simplifying exponential expressions, depending on the base and exponent.
• Not simplifying the expression completely: Make sure to simplify the expression completely, including any negative exponents or fractions.

Tips and Tricks

• Use the rule of exponents to simplify exponential expressions: The rule of exponents states that when multiplying two exponential expressions with the same base, you can multiply the bases and add the exponents.
• Use the rule of negative exponents to simplify exponential expressions with negative exponents: The rule of negative exponents states that an exponential expression with a negative exponent can be simplified by keeping the same base and adding the exponents.
• Use the rule of fractions to simplify exponential expressions with fractions as the base: The rule of fractions states that when multiplying two exponential expressions with the same base, you can multiply the bases and add the exponents.

Conclusion

In this article, we have answered some of the most frequently asked questions about exponential expressions. We have covered topics such as simplifying exponential expressions with the same base, using the rule of negative exponents, and simplifying exponential expressions with fractions as the base. By following the rules and tips outlined in this article, you can simplify exponential expressions with ease and gain a deeper understanding of exponential notation.