Simplify The Expression:${ \left(3 3\right) 2 }$ { \left(3^3\right)^2 = \square \} (Simplify Your Answer. Type An Integer Or A Fraction.)

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Understanding the Problem

When dealing with exponents, it's essential to understand the rules that govern their behavior. In this case, we're given the expression $\left(3^3\right)^2$ and asked to simplify it. To approach this problem, we need to recall the rule of exponents that states when we have a power raised to another power, we multiply the exponents.

Applying the Rule of Exponents

The rule of exponents states that for any numbers $a$, $b$, and $c$, we have:

$$\left(a^b\right)^c = a^{b \cdot c}$$

Using this rule, we can simplify the expression $\left(3^3\right)^2$ by multiplying the exponents:

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

Simplifying the Expression

Now that we have the expression in the form $3^{3 \cdot 2}$, we can simplify it further by evaluating the exponent:

$$3^{3 \cdot 2} = 3^6$$

Evaluating the Expression

To evaluate the expression $3^6$, we need to calculate the value of $3$ raised to the power of $6$. This can be done by multiplying $3$ by itself six times:

$$3^6 = 3 \cdot 3 \cdot 3 \cdot 3 \cdot 3 \cdot 3$$

Calculating the Value

Now, let's calculate the value of $3^6$:

$$3^6 = 3 \cdot 3 \cdot 3 \cdot 3 \cdot 3 \cdot 3 = 729$$

Conclusion

In conclusion, the expression $\left(3^3\right)^2$ simplifies to $729$. This is because we applied the rule of exponents to multiply the exponents, and then evaluated the resulting expression to calculate its value.

Additional Examples

To further illustrate the concept of simplifying expressions with exponents, let's consider a few additional examples:

Example 1

Simplify the expression $\left(2^4\right)^3$.

Using the rule of exponents, we have:

$$\left(2^4\right)^3 = 2^{4 \cdot 3} = 2^{12}$$

Evaluating the expression, we get:

$$2^{12} = 4096$$

Example 2

Simplify the expression $\left(5^2\right)^4$.

Using the rule of exponents, we have:

$$\left(5^2\right)^4 = 5^{2 \cdot 4} = 5^8$$

Evaluating the expression, we get:

$$5^8 = 390625$$

Final Thoughts

In this article, we simplified the expression $\left(3^3\right)^2$ using the rule of exponents. We also provided additional examples to illustrate the concept of simplifying expressions with exponents. By understanding and applying the rule of exponents, we can simplify complex expressions and evaluate their values.

Frequently Asked Questions

• Q: What is the rule of exponents? A: The rule of exponents states that for any numbers $a$, $b$, and $c$, we have: $\left(a^b\right)^c = a^{b \cdot c}$.
• Q: How do I simplify an expression with exponents? A: To simplify an expression with exponents, apply the rule of exponents by multiplying the exponents, and then evaluate the resulting expression.
• Q: What is the value of $3^6$? A: The value of $3^6$ is $729$.

References

• [1] "Exponents and Powers" by Math Open Reference
• [2] "Rules of Exponents" by Khan Academy

Related Articles

• Simplifying Expressions with Fractions
• Evaluating Expressions with Variables
  

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Frequently Asked Questions

Q: What is the rule of exponents?

A: The rule of exponents states that for any numbers $a$, $b$, and $c$, we have: $\left(a^b\right)^c = a^{b \cdot c}$. This rule allows us to simplify expressions with exponents by multiplying the exponents.

Q: How do I simplify an expression with exponents?

A: To simplify an expression with exponents, apply the rule of exponents by multiplying the exponents, and then evaluate the resulting expression. For example, to simplify the expression $\left(3^3\right)^2$, we would multiply the exponents: $\left(3^3\right)^2 = 3^{3 \cdot 2} = 3^6$.

Q: What is the value of $3^6$?

A: The value of $3^6$ is $729$. This is because we can calculate the value of $3^6$ by multiplying $3$ by itself six times: $3 \cdot 3 \cdot 3 \cdot 3 \cdot 3 \cdot 3 = 729$.

Q: Can I simplify expressions with negative exponents?

A: Yes, we can simplify expressions with negative exponents using the rule of exponents. For example, to simplify the expression $\left(2^{-3}\right)^2$, we would multiply the exponents: $\left(2^{-3}\right)^2 = 2^{-3 \cdot 2} = 2^{-6}$. This can be rewritten as $\frac{1}{2^6}$.

Q: How do I simplify expressions with fractional exponents?

A: To simplify expressions with fractional exponents, we can use the rule of exponents. For example, to simplify the expression $\left(2^{\frac{1}{2}}\right)^3$, we would multiply the exponents: $\left(2^{\frac{1}{2}}\right)^3 = 2^{\frac{1}{2} \cdot 3} = 2^{\frac{3}{2}}$.

Q: Can I simplify expressions with variables as exponents?

A: Yes, we can simplify expressions with variables as exponents using the rule of exponents. For example, to simplify the expression $\left(x^2\right)^3$, we would multiply the exponents: $\left(x^2\right)^3 = x^{2 \cdot 3} = x^6$.

Q: How do I evaluate expressions with exponents?

A: To evaluate expressions with exponents, we need to calculate the value of the expression. For example, to evaluate the expression $2^5$, we would multiply $2$ by itself five times: $2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 = 32$.

Q: Can I simplify expressions with multiple exponents?

A: Yes, we can simplify expressions with multiple exponents using the rule of exponents. For example, to simplify the expression $\left(2^3\right)^4 \cdot \left(3^2\right)^3$, we would multiply the exponents: $\left(2^3\right)^4 \cdot \left(3^2\right)^3 = 2^{3 \cdot 4} \cdot 3^{2 \cdot 3} = 2^{12} \cdot 3^6$.

Additional Resources

• Exponents and Powers
• Rules of Exponents
• Simplifying Expressions with Fractions
• Evaluating Expressions with Variables

Related Articles

• Simplifying Expressions with Fractions
• Evaluating Expressions with Variables
• Simplifying Expressions with Decimals
• Evaluating Expressions with Exponents