Simplify The Expression:$\[ \frac{2^0 A^2}{6^1} \\]A. 0B. \[$a^2\$\]C. \[$\frac{a^2}{6}\$\]D. \[$\frac{a^2}{3}\$\]

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Understanding the Expression

The given expression is $\frac{2^0 a^2}{6^1}$. To simplify this expression, we need to understand the properties of exponents and fractions. The expression consists of two terms: $2^0$ and $a^2$, which are divided by $6^1$.

Simplifying Exponents

Let's start by simplifying the exponents in the expression. The exponent $2^0$ represents $2$ raised to the power of $0$. According to the properties of exponents, any number raised to the power of $0$ is equal to $1$. Therefore, $2^0 = 1$.

Simplifying the Fraction

Now that we have simplified the exponent, let's focus on the fraction $\frac{a^2}{6^1}$. The numerator is $a^2$, and the denominator is $6^1$. To simplify the fraction, we need to find the greatest common divisor (GCD) of the numerator and the denominator.

Finding the Greatest Common Divisor (GCD)

The GCD of $a^2$ and $6^1$ is $1$, since $a$ is a variable and $6$ is a constant. Therefore, the fraction $\frac{a^2}{6^1}$ cannot be simplified further.

Simplifying the Expression

Now that we have simplified the exponents and the fraction, let's combine the two terms. The expression becomes $\frac{1 \cdot a^2}{6^1}$. Since $1$ is a multiplicative identity, we can cancel it out, leaving us with $\frac{a^2}{6}$.

Evaluating the Options

Now that we have simplified the expression, let's evaluate the options:

A. $0$ - This option is incorrect, since the expression is not equal to $0$.

B. $a^2$ - This option is incorrect, since the expression is not equal to $a^2$.

C. $\frac{a^2}{6}$ - This option is correct, since the simplified expression is indeed $\frac{a^2}{6}$.

D. $\frac{a^2}{3}$ - This option is incorrect, since the expression is not equal to $\frac{a^2}{3}$.

The final answer is $\boxed{\frac{a^2}{6}}$.

Conclusion

In this article, we simplified the expression $\frac{2^0 a^2}{6^1}$ by understanding the properties of exponents and fractions. We found that the expression simplifies to $\frac{a^2}{6}$. We then evaluated the options and found that the correct answer is $\boxed{\frac{a^2}{6}}$.

Frequently Asked Questions

• What is the value of $2^0$? $2^0 = 1$
• What is the greatest common divisor (GCD) of $a^2$ and $6^1$? The GCD of $a^2$ and $6^1$ is $1$.
• What is the simplified expression? The simplified expression is $\frac{a^2}{6}$.

Related Topics

• Simplifying expressions with exponents
• Understanding the properties of fractions
• Evaluating options in math problems

References

• [1] Khan Academy. (n.d.). Exponents and Exponential Functions. Retrieved from https://www.khanacademy.org/math/algebra/x2f5f7d/x2f5f7d/exponents-and-exponential-functions
• [2] Math Open Reference. (n.d.). Fractions. Retrieved from https://www.mathopenref.com/fractions.html
  

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Frequently Asked Questions

Q: What is the value of $2^0$?

A: The value of $2^0$ is $1$. According to the properties of exponents, any number raised to the power of $0$ is equal to $1$.

Q: What is the greatest common divisor (GCD) of $a^2$ and $6^1$?

A: The GCD of $a^2$ and $6^1$ is $1$. Since $a$ is a variable and $6$ is a constant, there is no common factor between $a^2$ and $6^1$.

Q: What is the simplified expression?

A: The simplified expression is $\frac{a^2}{6}$. We simplified the expression by understanding the properties of exponents and fractions.

Q: Why is option A incorrect?

A: Option A is incorrect because the expression is not equal to $0$. The expression $\frac{2^0 a^2}{6^1}$ is equal to $\frac{a^2}{6}$, not $0$.

Q: Why is option B incorrect?

A: Option B is incorrect because the expression is not equal to $a^2$. The expression $\frac{2^0 a^2}{6^1}$ is equal to $\frac{a^2}{6}$, not $a^2$.

Q: Why is option D incorrect?

A: Option D is incorrect because the expression is not equal to $\frac{a^2}{3}$. The expression $\frac{2^0 a^2}{6^1}$ is equal to $\frac{a^2}{6}$, not $\frac{a^2}{3}$.

Q: What is the final answer?

A: The final answer is $\boxed{\frac{a^2}{6}}$.

Additional Questions and Answers

Q: What is the property of exponents that states any number raised to the power of $0$ is equal to $1$?

A: The property of exponents that states any number raised to the power of $0$ is equal to $1$ is known as the zero exponent rule.

Q: What is the greatest common divisor (GCD) of two numbers?

A: The greatest common divisor (GCD) of two numbers is the largest number that divides both numbers without leaving a remainder.

Q: How do you simplify an expression with exponents and fractions?

A: To simplify an expression with exponents and fractions, you need to understand the properties of exponents and fractions. You can simplify the expression by canceling out common factors and using the zero exponent rule.

Related Topics

• Simplifying expressions with exponents
• Understanding the properties of fractions
• Evaluating options in math problems
• Greatest common divisor (GCD)
• Zero exponent rule

References

• [1] Khan Academy. (n.d.). Exponents and Exponential Functions. Retrieved from https://www.khanacademy.org/math/algebra/x2f5f7d/x2f5f7d/exponents-and-exponential-functions
• [2] Math Open Reference. (n.d.). Fractions. Retrieved from https://www.mathopenref.com/fractions.html
• [3] Purplemath. (n.d.). Greatest Common Divisor (GCD). Retrieved from https://www.purplemath.com/modules/gcd.html
• [4] Mathway. (n.d.). Zero Exponent Rule. Retrieved from https://www.mathway.com/subjects/Algebra/Zero-Exponent-Rule