Question

rewrite using a single positive exponent. \\(\frac{7^4}{7}\\)

Explanation:

Step1: Recall exponent rule

The rule for dividing exponents with the same base is \( \frac{a^m}{a^n}=a^{m - n} \) when \( a
eq0 \). Here, the base \( a = 7 \), \( m = 4 \), \( n = 7 \).

Step2: Apply the rule

\( \frac{7^4}{7^7}=7^{4 - 7} \)

Step3: Simplify the exponent

\( 4-7=- 3 \), so we have \( 7^{-3} \). But we need a single positive exponent. Recall that \( a^{-n}=\frac{1}{a^n} \), so \( 7^{-3}=\frac{1}{7^3} \), or we can also use the rule in reverse: \( \frac{a^m}{a^n}=\frac{1}{a^{n - m}} \) when \( n>m \). So \( \frac{7^4}{7^7}=\frac{1}{7^{7 - 4}}=\frac{1}{7^3} \) (or if we consider the other way, but the problem says "rewrite using a single positive exponent", so we can express it as \( \frac{1}{7^3} \) or recognize that maybe there was a typo and if it's \( \frac{7^7}{7^4} \) then it's \( 7^{3} \), but based on the given \( \frac{7^4}{7^7} \), using the rule for positive exponent, we can write it as \( \frac{1}{7^{3}} \), but also, another way: the negative exponent rule, but to have a single positive exponent in the denominator (or numerator). Wait, the problem says "rewrite using a single positive exponent". So using the exponent rule \( \frac{a^m}{a^n}=a^{m - n} \), but if \( m - n \) is negative, we can write it as \( \frac{1}{a^{n - m}} \), which has a positive exponent in the denominator. So for \( \frac{7^4}{7^7} \), \( n - m=7 - 4 = 3 \), so it's \( \frac{1}{7^3} \), but also, if we consider that maybe the original was \( \frac{7^7}{7^4} \), then it's \( 7^{3} \). Wait, maybe the user made a typo, but based on the given \( \frac{7^4}{7^7} \), let's proceed. So applying \( \frac{a^m}{a^n}=a^{m - n} \), we get \( 7^{-3} \), but to make the exponent positive, we use \( a^{-n}=\frac{1}{a^n} \), so \( 7^{-3}=\frac{1}{7^3} \), which has a single positive exponent \( 3 \) in the denominator.

Answer:

\( \frac{1}{7^3} \) (or if the original was \( \frac{7^7}{7^4} \), then \( 7^3 \); but based on the given \( \frac{7^4}{7^7} \), the answer is \( \frac{1}{7^3} \))