Evaluate The Expression $\left(2^2 \cdot 2^3\right)^3$.Your Answer Is $\square$.

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Introduction

In mathematics, the concept of exponents is crucial in simplifying complex expressions. When dealing with exponents, it's essential to understand the rules of exponentiation, including the product of powers rule, the power of a power rule, and the power of a product rule. In this article, we will evaluate the expression $\left(2^2 \cdot 2^3\right)^3$ using these rules.

Understanding Exponents

Exponents are a shorthand way of representing repeated multiplication. For example, $2^3$ means $2$ multiplied by itself $3$ times, which is equal to $8$. When dealing with exponents, it's essential to understand the rules of exponentiation.

Product of Powers Rule

The product of powers rule states that when multiplying two powers with the same base, we add the exponents. For example, $2^3 \cdot 2^4 = 2^{3+4} = 2^7$. This rule is essential in simplifying complex expressions.

Power of a Power Rule

The power of a power rule states that when raising a power to another power, we multiply the exponents. For example, $(2^3)^4 = 2^{3 \cdot 4} = 2^{12}$. This rule is also essential in simplifying complex expressions.

Power of a Product Rule

The power of a product rule states that when raising a product to a power, we raise each factor to that power. For example, $(2 \cdot 3)^4 = 2^4 \cdot 3^4$. This rule is essential in simplifying complex expressions.

Evaluating the Expression

Now that we understand the rules of exponentiation, let's evaluate the expression $\left(2^2 \cdot 2^3\right)^3$. To do this, we will apply the product of powers rule to simplify the expression inside the parentheses.

Simplifying the Expression Inside the Parentheses

Using the product of powers rule, we can simplify the expression inside the parentheses as follows:

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

Applying the Power of a Power Rule

Now that we have simplified the expression inside the parentheses, we can apply the power of a power rule to evaluate the entire expression.

$$(2^5)^3 = 2^{5 \cdot 3} = 2^{15}$$

Conclusion

In conclusion, the expression $\left(2^2 \cdot 2^3\right)^3$ can be evaluated using the rules of exponentiation. By applying the product of powers rule and the power of a power rule, we can simplify the expression and arrive at the final answer.

Final Answer

The final answer is $\boxed{32768}$.

Additional Examples

Here are some additional examples of evaluating expressions using the rules of exponentiation:

• $\left(3^2 \cdot 3^4\right)^3 = 3^{2+4} \cdot 3^3 = 3^6 \cdot 3^3 = 3^{6+3} = 3^9$
• $\left(4^3 \cdot 4^2\right)^4 = 4^{3+2} \cdot 4^4 = 4^5 \cdot 4^4 = 4^{5+4} = 4^9$
• $\left(5^4 \cdot 5^3\right)^5 = 5^{4+3} \cdot 5^5 = 5^7 \cdot 5^5 = 5^{7+5} = 5^{12}$

These examples demonstrate the application of the product of powers rule and the power of a power rule in evaluating complex expressions.

Tips and Tricks

Here are some tips and tricks for evaluating expressions using the rules of exponentiation:

• Always start by simplifying the expression inside the parentheses using the product of powers rule.
• Then, apply the power of a power rule to evaluate the entire expression.
• Make sure to add the exponents when multiplying two powers with the same base.
• Make sure to multiply the exponents when raising a power to another power.

By following these tips and tricks, you can simplify complex expressions and arrive at the final answer.

Common Mistakes

Here are some common mistakes to avoid when evaluating expressions using the rules of exponentiation:

• Failing to simplify the expression inside the parentheses.
• Failing to apply the power of a power rule.
• Adding the exponents when multiplying two powers with different bases.
• Multiplying the exponents when raising a power to another power.

By avoiding these common mistakes, you can ensure that your calculations are accurate and your final answer is correct.

Conclusion

In conclusion, evaluating expressions using the rules of exponentiation is a crucial skill in mathematics. By understanding the product of powers rule, the power of a power rule, and the power of a product rule, you can simplify complex expressions and arrive at the final answer. Remember to always start by simplifying the expression inside the parentheses, then apply the power of a power rule, and finally, add the exponents when multiplying two powers with the same base. By following these steps, you can become proficient in evaluating expressions using the rules of exponentiation.

Introduction

In our previous article, we discussed how to evaluate expressions using the rules of exponentiation. We covered the product of powers rule, the power of a power rule, and the power of a product rule. In this article, we will answer some frequently asked questions about evaluating expressions using the rules of exponentiation.

Q&A

Q: What is the product of powers rule?

A: The product of powers rule states that when multiplying two powers with the same base, we add the exponents. For example, $2^3 \cdot 2^4 = 2^{3+4} = 2^7$.

Q: What is the power of a power rule?

A: The power of a power rule states that when raising a power to another power, we multiply the exponents. For example, $(2^3)^4 = 2^{3 \cdot 4} = 2^{12}$.

Q: What is the power of a product rule?

A: The power of a product rule states that when raising a product to a power, we raise each factor to that power. For example, $(2 \cdot 3)^4 = 2^4 \cdot 3^4$.

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

A: To simplify an expression with multiple exponents, start by simplifying the expression inside the parentheses using the product of powers rule. Then, apply the power of a power rule to evaluate the entire expression.

Q: What is the difference between adding and multiplying exponents?

A: When multiplying two powers with the same base, we add the exponents. For example, $2^3 \cdot 2^4 = 2^{3+4} = 2^7$. When raising a power to another power, we multiply the exponents. For example, $(2^3)^4 = 2^{3 \cdot 4} = 2^{12}$.

Q: Can I simplify an expression with a negative exponent?

A: Yes, you can simplify an expression with a negative exponent. To do this, rewrite the expression with a positive exponent by moving the base to the other side of the fraction bar. For example, $2^{-3} = \frac{1}{2^3} = \frac{1}{8}$.

Q: How do I evaluate an expression with a zero exponent?

A: An expression with a zero exponent is equal to 1. For example, $2^0 = 1$.

Q: Can I simplify an expression with a fractional exponent?

A: Yes, you can simplify an expression with a fractional exponent. To do this, rewrite the expression as a product of two powers. For example, $2^{\frac{1}{2}} = \sqrt{2}$.

Tips and Tricks

Here are some tips and tricks for evaluating expressions using the rules of exponentiation:

• Always start by simplifying the expression inside the parentheses using the product of powers rule.
• Then, apply the power of a power rule to evaluate the entire expression.
• Make sure to add the exponents when multiplying two powers with the same base.
• Make sure to multiply the exponents when raising a power to another power.
• Be careful when simplifying expressions with negative exponents or fractional exponents.

Common Mistakes

Here are some common mistakes to avoid when evaluating expressions using the rules of exponentiation:

• Failing to simplify the expression inside the parentheses.
• Failing to apply the power of a power rule.
• Adding the exponents when multiplying two powers with different bases.
• Multiplying the exponents when raising a power to another power.
• Failing to rewrite expressions with negative exponents or fractional exponents.

Conclusion

In conclusion, evaluating expressions using the rules of exponentiation is a crucial skill in mathematics. By understanding the product of powers rule, the power of a power rule, and the power of a product rule, you can simplify complex expressions and arrive at the final answer. Remember to always start by simplifying the expression inside the parentheses, then apply the power of a power rule, and finally, add the exponents when multiplying two powers with the same base. By following these steps, you can become proficient in evaluating expressions using the rules of exponentiation.