Question

rewrite using a single positive exponent.\\(\frac{5^{-3}}{5^{6}}\\)

Explanation:

Step1: Apply exponent division rule

When dividing exponents with the same base, subtract the exponents: \( \frac{a^m}{a^n} = a^{m - n} \). Here, \( a = 5 \), \( m = -3 \), \( n = 6 \). So we get \( 5^{-3 - 6} \).

Step2: Simplify the exponent

Calculate \( -3 - 6 = -9 \), so the expression becomes \( 5^{-9} \). But we need a positive exponent. Recall that \( a^{-n} = \frac{1}{a^n} \), but we can also use the rule in reverse: \( \frac{a^m}{a^n}=a^{m - n}\), and if we want positive exponent, we can rewrite as \( \frac{1}{a^{n - m}} \) when \( m < n \). Alternatively, since we have \( 5^{-3}/5^{6}=5^{-3 - 6}=5^{-9}=\frac{1}{5^{9}} \), but wait, no—wait, the rule for dividing with same base is \( a^m\div a^n=a^{m - n} \). So \( 5^{-3} \div 5^{6}=5^{-3 - 6}=5^{-9} \), and to make the exponent positive, we use \( a^{-k}=\frac{1}{a^{k}} \), but the problem says "rewrite using a single positive exponent"—wait, maybe I made a mistake. Wait, no: actually, when you have \( \frac{a^m}{a^n} \), it's \( a^{m - n} \). So \( 5^{-3} / 5^{6}=5^{-3 - 6}=5^{-9} \), and \( 5^{-9}=\frac{1}{5^{9}} \), but \( \frac{1}{5^{9}} \) is \( 5^{-9} \), but we need a single positive exponent. Wait, maybe the problem allows expressing it as \( \frac{1}{5^{9}} \), but \( \frac{1}{5^{9}} \) is \( 5^{-9} \), but no—wait, no, the rule for negative exponents: \( a^{-n}=\frac{1}{a^n} \), so to write with positive exponent, we can write \( \frac{1}{5^{9}} \), but the problem says "rewrite using a single positive exponent"—wait, maybe I messed up the sign. Wait, \( 5^{-3} / 5^{6}=5^{-3 - 6}=5^{-9}=\frac{1}{5^{9}} \), and \( \frac{1}{5^{9}} \) has a positive exponent in the denominator, but the problem says "a single positive exponent"—maybe it's acceptable to write \( 5^{-9} \) as \( \frac{1}{5^{9}} \), but \( \frac{1}{5^{9}} \) is \( 5^{-9} \), but the exponent in \( 5^{9} \) is positive. Wait, the problem says "rewrite using a single positive exponent"—so we can write it as \( \frac{1}{5^{9}} \), but \( \frac{1}{5^{9}} \) is \( 5^{-9} \), but the exponent 9 is positive. Wait, maybe the question is to have the base with a positive exponent, so \( 5^{-9}=\frac{1}{5^{9}} \), which is \( 5^{9} \) in the denominator, but as a single term with positive exponent, it's \( 5^{-9} \) is not positive, but \( \frac{1}{5^{9}} \) has \( 5^{9} \) with positive exponent. Wait, maybe I made a mistake in the exponent subtraction. Let's recheck: \( \frac{5^{-3}}{5^{6}} = 5^{-3 - 6} = 5^{-9} \), and using the negative exponent rule, \( 5^{-9} = \frac{1}{5^{9}} \), so \( \frac{1}{5^{9}} \) is written with a single positive exponent (9) in the denominator. So the answer is \( \frac{1}{5^{9}} \) or \( 5^{-9} \), but since we need positive exponent, it's \( \frac{1}{5^{9}} \) which is \( 5^{9} \) in the denominator, so as a single term with positive exponent, it's \( 5^{-9} \) is not positive, but \( \frac{1}{5^{9}} \) has \( 5^{9} \) with positive exponent. So the correct rewrite with single positive exponent is \( \frac{1}{5^{9}} \), which is \( 5^{-9} \) but expressed with positive exponent in the denominator.

Answer:

\( \frac{1}{5^{9}} \) (or \( 5^{-9} \) is incorrect for positive exponent, so the correct is \( \frac{1}{5^{9}} \) with positive exponent 9 in the denominator)