Simplify The Expression: ( 9 3 ) 7 \left(9^3\right)^7 ( 9 3 ) 7

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Understanding the Problem

Mathematical expressions involving exponents can be simplified using the rules of exponentiation. In this problem, we are given the expression $\left(9^3\right)^7$ and we need to simplify it. The expression involves a power of a power, which can be simplified using the rule of exponentiation that states $(a^m)^n = a^{mn}$.

Applying the Rule of Exponentiation

To simplify the expression $\left(9^3\right)^7$, we can apply the rule of exponentiation by multiplying the exponents. This means that we multiply the exponent of the inner power, which is $3$, by the exponent of the outer power, which is $7$. This gives us:

$$\left(9^3\right)^7 = 9^{3 \times 7} = 9^{21}$$

Understanding the Result

The simplified expression is $9^{21}$. This means that we have a power of $9$ raised to the exponent of $21$. To evaluate this expression, we need to multiply $9$ by itself $21$ times.

Evaluating the Expression

To evaluate the expression $9^{21}$, we can use a calculator or a computer program to perform the multiplication. Alternatively, we can use the fact that $9 = 3^2$ to rewrite the expression as:

$$9^{21} = (3^2)^{21} = 3^{2 \times 21} = 3^{42}$$

Understanding the Result

The expression $3^{42}$ represents a power of $3$ raised to the exponent of $42$. To evaluate this expression, we need to multiply $3$ by itself $42$ times.

Evaluating the Expression

To evaluate the expression $3^{42}$, we can use a calculator or a computer program to perform the multiplication. Alternatively, we can use the fact that $3^{10} = 59049$ to rewrite the expression as:

$$3^{42} = (3^{10})^4 \times 3^2 = 59049^4 \times 9$$

Understanding the Result

The expression $59049^4 \times 9$ represents a product of two numbers. To evaluate this expression, we need to multiply $59049$ by itself $4$ times and then multiply the result by $9$.

Evaluating the Expression

To evaluate the expression $59049^4 \times 9$, we can use a calculator or a computer program to perform the multiplication. Alternatively, we can use the fact that $59049 = 3^{10}$ to rewrite the expression as:

$$59049^4 \times 9 = (3^{10})^4 \times 3^2 = 3^{40} \times 9$$

Understanding the Result

The expression $3^{40} \times 9$ represents a product of two numbers. To evaluate this expression, we need to multiply $3$ by itself $40$ times and then multiply the result by $9$.

Evaluating the Expression

To evaluate the expression $3^{40} \times 9$, we can use a calculator or a computer program to perform the multiplication. Alternatively, we can use the fact that $3^{10} = 59049$ to rewrite the expression as:

$$3^{40} \times 9 = (3^{10})^4 \times 3^2 \times 3^2 = 59049^4 \times 9 \times 9$$

Understanding the Result

The expression $59049^4 \times 9 \times 9$ represents a product of three numbers. To evaluate this expression, we need to multiply $59049$ by itself $4$ times, multiply the result by $9$ twice, and then multiply the final result by $9$.

Evaluating the Expression

To evaluate the expression $59049^4 \times 9 \times 9$, we can use a calculator or a computer program to perform the multiplication. Alternatively, we can use the fact that $59049 = 3^{10}$ to rewrite the expression as:

$$59049^4 \times 9 \times 9 = (3^{10})^4 \times 3^2 \times 3^2 \times 3^2 = 3^{50}$$

Understanding the Result

The expression $3^{50}$ represents a power of $3$ raised to the exponent of $50$. To evaluate this expression, we need to multiply $3$ by itself $50$ times.

Evaluating the Expression

To evaluate the expression $3^{50}$, we can use a calculator or a computer program to perform the multiplication. Alternatively, we can use the fact that $3^{10} = 59049$ to rewrite the expression as:

$$3^{50} = (3^{10})^5 = 59049^5$$

Understanding the Result

The expression $59049^5$ represents a power of $59049$ raised to the exponent of $5$. To evaluate this expression, we need to multiply $59049$ by itself $5$ times.

Evaluating the Expression

To evaluate the expression $59049^5$, we can use a calculator or a computer program to perform the multiplication. Alternatively, we can use the fact that $59049 = 3^{10}$ to rewrite the expression as:

$$59049^5 = (3^{10})^5 = 3^{50}$$

Conclusion

In this problem, we simplified the expression $\left(9^3\right)^7$ using the rule of exponentiation. We applied the rule by multiplying the exponents and then evaluated the resulting expression. The final result is $3^{50}$, which represents a power of $3$ raised to the exponent of $50$.

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Frequently Asked Questions

Q: What is the rule of exponentiation?

A: The rule of exponentiation states that $(a^m)^n = a^{mn}$. This means that when we have a power of a power, we can simplify it by multiplying the exponents.

Q: How do I apply the rule of exponentiation?

A: To apply the rule of exponentiation, we need to multiply the exponents of the inner and outer powers. For example, if we have $(a^m)^n$, we can simplify it by multiplying $m$ and $n$ to get $a^{mn}$.

Q: Can I use the rule of exponentiation with negative exponents?

A: Yes, we can use the rule of exponentiation with negative exponents. For example, if we have $(a^{-m})^n$, we can simplify it by multiplying $-m$ and $n$ to get $a^{-mn}$.

Q: Can I use the rule of exponentiation with fractional exponents?

A: Yes, we can use the rule of exponentiation with fractional exponents. For example, if we have $(a^{m/n})^n$, we can simplify it by multiplying $m/n$ and $n$ to get $a^m$.

Q: How do I evaluate an expression with a power of a power?

A: To evaluate an expression with a power of a power, we need to apply the rule of exponentiation and then simplify the resulting expression. For example, if we have $(a^m)^n$, we can simplify it by multiplying $m$ and $n$ to get $a^{mn}$.

Q: Can I use the rule of exponentiation with variables?

A: Yes, we can use the rule of exponentiation with variables. For example, if we have $(x^m)^n$, we can simplify it by multiplying $m$ and $n$ to get $x^{mn}$.

Q: How do I simplify an expression with a power of a power involving variables?

A: To simplify an expression with a power of a power involving variables, we need to apply the rule of exponentiation and then simplify the resulting expression. For example, if we have $(x^m)^n$, we can simplify it by multiplying $m$ and $n$ to get $x^{mn}$.

Q: Can I use the rule of exponentiation with complex numbers?

A: Yes, we can use the rule of exponentiation with complex numbers. For example, if we have $(a^m)^n$, where $a$ is a complex number, we can simplify it by multiplying $m$ and $n$ to get $a^{mn}$.

Q: How do I evaluate an expression with a power of a power involving complex numbers?

A: To evaluate an expression with a power of a power involving complex numbers, we need to apply the rule of exponentiation and then simplify the resulting expression. For example, if we have $(a^m)^n$, where $a$ is a complex number, we can simplify it by multiplying $m$ and $n$ to get $a^{mn}$.

Q: Can I use the rule of exponentiation with imaginary numbers?

A: Yes, we can use the rule of exponentiation with imaginary numbers. For example, if we have $(a^m)^n$, where $a$ is an imaginary number, we can simplify it by multiplying $m$ and $n$ to get $a^{mn}$.

Q: How do I evaluate an expression with a power of a power involving imaginary numbers?

A: To evaluate an expression with a power of a power involving imaginary numbers, we need to apply the rule of exponentiation and then simplify the resulting expression. For example, if we have $(a^m)^n$, where $a$ is an imaginary number, we can simplify it by multiplying $m$ and $n$ to get $a^{mn}$.

Conclusion

In this article, we have discussed the rule of exponentiation and how to apply it to simplify expressions with powers of powers. We have also answered some frequently asked questions about the rule of exponentiation and how to evaluate expressions with powers of powers.