What Is The Value Of ( 3 10 ) 3 \left(\frac{3}{10}\right)^3 ( 10 3 ​ ) 3 ?A. 3 1000 \frac{3}{1000} 1000 3 ​ B. 27 1000 \frac{27}{1000} 1000 27 ​ C. 9 10 \frac{9}{10} 10 9 ​ D. 27 10 \frac{27}{10} 10 27 ​

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Understanding Exponents and Fractions

When dealing with exponents and fractions, it's essential to understand the rules and operations involved. In this case, we're tasked with finding the value of $\left(\frac{3}{10}\right)^3$. To approach this problem, we need to recall the rule for raising a fraction to a power. When a fraction is raised to a power, both the numerator and the denominator are raised to that power.

Applying the Rule for Raising a Fraction to a Power

To find the value of $\left(\frac{3}{10}\right)^3$, we apply the rule by raising both the numerator and the denominator to the power of 3. This gives us:

$$\left(\frac{3}{10}\right)^3 = \frac{3^3}{10^3}$$

Calculating the Numerator and Denominator

Now that we have the expression $\frac{3^3}{10^3}$, we need to calculate the values of the numerator and the denominator. The numerator is $3^3$, which is equal to $3 \times 3 \times 3 = 27$. The denominator is $10^3$, which is equal to $10 \times 10 \times 10 = 1000$.

Simplifying the Expression

Now that we have the values of the numerator and the denominator, we can simplify the expression $\frac{3^3}{10^3}$. This gives us:

$$\frac{3^3}{10^3} = \frac{27}{1000}$$

Conclusion

In conclusion, the value of $\left(\frac{3}{10}\right)^3$ is $\frac{27}{1000}$. This is the correct answer among the options provided.

Comparison with Other Options

Let's compare our answer with the other options provided:

• Option A: $\frac{3}{1000}$ is incorrect because the numerator is not 3, but 27.
• Option B: $\frac{27}{1000}$ is correct, which is our answer.
• Option C: $\frac{9}{10}$ is incorrect because the denominator is not 10, but 1000.
• Option D: $\frac{27}{10}$ is incorrect because the denominator is not 10, but 1000.

Final Answer

The final answer is $\boxed{\frac{27}{1000}}$.

Q: What is the rule for raising a fraction to a power?

A: When a fraction is raised to a power, both the numerator and the denominator are raised to that power. This means that if we have a fraction $\frac{a}{b}$ and we raise it to a power $n$, we get $\frac{a^n}{b^n}$.

Q: How do I calculate the value of a fraction raised to a power?

A: To calculate the value of a fraction raised to a power, you need to raise both the numerator and the denominator to that power. For example, if we have $\left(\frac{3}{10}\right)^3$, we would calculate the value as $\frac{3^3}{10^3}$.

Q: What is the difference between a fraction and an exponent?

A: A fraction is a way of expressing a part of a whole, while an exponent is a way of expressing repeated multiplication. For example, the fraction $\frac{1}{2}$ represents one half of a whole, while the exponent $2^3$ represents $2$ multiplied by itself $3$ times.

Q: Can I simplify a fraction raised to a power?

A: Yes, you can simplify a fraction raised to a power by simplifying the numerator and the denominator separately. For example, if we have $\frac{2^3}{2^3}$, we can simplify it to $1$ because the numerator and the denominator are the same.

Q: How do I handle negative exponents?

A: When dealing with negative exponents, you can rewrite the fraction as a positive exponent by flipping the fraction. For example, if we have $\frac{1}{2^{-3}}$, we can rewrite it as $2^3$.

Q: Can I use exponents with decimals?

A: Yes, you can use exponents with decimals. For example, if we have $\left(\frac{0.5}{2}\right)^3$, we can calculate the value as $\frac{(0.5)^3}{2^3}$.

Q: How do I handle fractions with variables in the exponent?

A: When dealing with fractions with variables in the exponent, you can use the same rules as with numerical exponents. For example, if we have $\left(\frac{x}{y}\right)^3$, we can calculate the value as $\frac{x^3}{y^3}$.

Q: Can I use exponents with fractions in the denominator?

A: Yes, you can use exponents with fractions in the denominator. For example, if we have $\frac{1}{\left(\frac{1}{2}\right)^3}$, we can calculate the value as $\frac{1}{\frac{1^3}{2^3}}$.

Q: How do I handle complex fractions with exponents?

A: When dealing with complex fractions with exponents, you can use the same rules as with simple fractions. For example, if we have $\frac{\left(\frac{1}{2}\right)^3}{\left(\frac{1}{3}\right)^3}$, we can calculate the value as $\frac{\frac{1^3}{2^3}}{\frac{1^3}{3^3}}$.

Q: Can I use exponents with mixed numbers?

A: Yes, you can use exponents with mixed numbers. For example, if we have $\left(\frac{2\frac{1}{2}}{3}\right)^3$, we can calculate the value as $\frac{(2\frac{1}{2})^3}{3^3}$.

Q: How do I handle fractions with exponents in the numerator and denominator?

A: When dealing with fractions with exponents in the numerator and denominator, you can use the same rules as with simple fractions. For example, if we have $\frac{\left(\frac{1}{2}\right)^3}{\left(\frac{1}{3}\right)^3}$, we can calculate the value as $\frac{\frac{1^3}{2^3}}{\frac{1^3}{3^3}}$.

Q: Can I use exponents with fractions in the numerator and denominator?

A: Yes, you can use exponents with fractions in the numerator and denominator. For example, if we have $\frac{\left(\frac{1}{2}\right)^3}{\left(\frac{1}{3}\right)^3}$, we can calculate the value as $\frac{\frac{1^3}{2^3}}{\frac{1^3}{3^3}}$.

Q: How do I handle complex fractions with exponents and variables?

A: When dealing with complex fractions with exponents and variables, you can use the same rules as with simple fractions. For example, if we have $\frac{\left(\frac{x}{y}\right)^3}{\left(\frac{a}{b}\right)^3}$, we can calculate the value as $\frac{\frac{x^3}{y^3}}{\frac{a^3}{b^3}}$.

Q: Can I use exponents with fractions in the numerator and denominator with variables?

A: Yes, you can use exponents with fractions in the numerator and denominator with variables. For example, if we have $\frac{\left(\frac{x}{y}\right)^3}{\left(\frac{a}{b}\right)^3}$, we can calculate the value as $\frac{\frac{x^3}{y^3}}{\frac{a^3}{b^3}}$.

Q: How do I handle fractions with exponents in the numerator and denominator with decimals?

A: When dealing with fractions with exponents in the numerator and denominator with decimals, you can use the same rules as with simple fractions. For example, if we have $\frac{\left(\frac{0.5}{2}\right)^3}{\left(\frac{1}{3}\right)^3}$, we can calculate the value as $\frac{\frac{(0.5)^3}{2^3}}{\frac{1^3}{3^3}}$.

Q: Can I use exponents with fractions in the numerator and denominator with decimals?

A: Yes, you can use exponents with fractions in the numerator and denominator with decimals. For example, if we have $\frac{\left(\frac{0.5}{2}\right)^3}{\left(\frac{1}{3}\right)^3}$, we can calculate the value as $\frac{\frac{(0.5)^3}{2^3}}{\frac{1^3}{3^3}}$.

Q: How do I handle complex fractions with exponents, variables, and decimals?

A: When dealing with complex fractions with exponents, variables, and decimals, you can use the same rules as with simple fractions. For example, if we have $\frac{\left(\frac{x}{y}\right)^3}{\left(\frac{a}{b}\right)^3}$, we can calculate the value as $\frac{\frac{x^3}{y^3}}{\frac{a^3}{b^3}}$.

Q: Can I use exponents with fractions in the numerator and denominator with variables and decimals?

A: Yes, you can use exponents with fractions in the numerator and denominator with variables and decimals. For example, if we have $\frac{\left(\frac{x}{y}\right)^3}{\left(\frac{a}{b}\right)^3}$, we can calculate the value as $\frac{\frac{x^3}{y^3}}{\frac{a^3}{b^3}}$.

Q: How do I handle fractions with exponents in the numerator and denominator with mixed numbers?

A: When dealing with fractions with exponents in the numerator and denominator with mixed numbers, you can use the same rules as with simple fractions. For example, if we have $\frac{\left(\frac{2\frac{1}{2}}{3}\right)^3}{\left(\frac{1}{3}\right)^3}$, we can calculate the value as $\frac{\frac{(2\frac{1}{2})^3}{3^3}}{\frac{1^3}{3^3}}$.

Q: Can I use exponents with fractions in the numerator and denominator with mixed numbers?

A: Yes, you can use exponents with fractions in the numerator and denominator with mixed numbers. For example, if we have $\frac{\left(\frac{2\frac{1}{2}}{3}\right)^3}{\left(\frac{1}{3}\right)^3}$, we can calculate the value as $\frac{\frac{(2\frac{1}{2})^3}{3^3}}{\frac{1^3}{3^3}}$.

Q: How do I handle complex fractions with exponents, variables, decimals, and mixed numbers?

A: When dealing with complex fractions with exponents, variables, decimals, and mixed numbers, you can use the same rules as with simple fractions. For example, if we have $\frac{\left(\frac{x}{y}\right)^3}{\left(\frac{a}{b}\right)^3}$, we can calculate the value as $\frac{\frac{x^3}{y^3}}{\frac{a^3}{b^3}}$.

Q: Can I use exponents with fractions in the numerator and denominator with variables, decimals, and mixed numbers?

A: Yes, you can use exponents with fractions in the numerator and denominator with variables, decimals, and mixed