Question

which expression is equivalent to \\(\frac{5^{12}}{5^6}\\)? \
a \\(5^6\\) \
b \\(5^2\\) \
c \\(5^{18}\\) \
d \\(5^{72}\\)

Explanation:

Step1: Recall exponent rule for division

When dividing exponents with the same base, we use the rule \( \frac{a^m}{a^n}=a^{m - n} \), where \( a = 5 \), \( m = 12 \), and \( n = 6 \).

Step2: Apply the rule

Substitute the values into the rule: \( \frac{5^{12}}{5^{6}}=5^{12 - 6} \).

Step3: Calculate the exponent

Simplify the exponent: \( 12-6 = 6 \)? Wait, no, wait, wait, no, wait, the original problem is \( \frac{5^{12}}{5^{6}} \)? Wait, no, looking back, the problem is \( 5^{12} \) divided by \( 5^{6} \)? Wait, no, the user's problem: "Which expression is equivalent to \( \boldsymbol{5^{12}} \) \( \boldsymbol{5^{6}} \)? Wait, no, the LaTeX is wrong. Wait, the image shows \( \frac{5^{12}}{5^{6}} \). So the correct rule is \( \frac{a^m}{a^n}=a^{m - n} \). So \( m = 12 \), \( n = 6 \), so \( 12 - 6 = 6 \)? Wait, no, that can't be. Wait, no, the options: A is \( 5^6 \), B is \( 5^2 \), C is \( 5^{18} \), D is \( 5^{72} \). Wait, maybe I misread the exponent. Wait, maybe the numerator is \( 5^{12} \) and denominator is \( 5^6 \)? Wait, no, maybe the problem is \( 5^{12} \times 5^6 \)? Wait, the user's image: "Which expression is equivalent to \( 5^{12} \) \( 5^6 \)?" Wait, maybe it's a multiplication? Wait, no, the options: if it's division, \( 5^{12}/5^6 = 5^{6} \), which is option A. But let's check again.

Wait, the problem as per the image: "Which expression is equivalent to \( \frac{5^{12}}{5^{6}} \)?" So using the exponent rule \( \frac{a^m}{a^n}=a^{m - n} \), so \( 5^{12 - 6}=5^6 \). So the correct option is A.

Wait, but let's confirm:

Exponent rule for division: when dividing two powers with the same base, subtract the exponents. So \( 5^{12} \div 5^6 = 5^{12 - 6}=5^6 \), which is option A.

Answer:

A. \( 5^6 \)