Question

properties of exponents
#1 \\(\frac{8^4}{8^7}\\) #2 \\((5^6)(5^3)\\) #3 \\((7^6)^8\\) #4 \\((x^6)^3\\)
#5 \\(\frac{a^5}{a^5}\\) #6 \\((7^{-7})(7^{12})\\) #7 \\(\frac{8^4}{8^3} \cdot 8^9\\) #8 \\((3^6 \cdot 3^2)^0\\)
#9 \\(\frac{10^7}{10^5} \cdot 10^{-5}\\) #10 \\((5^3)(5^4)(5)\\) #11 \\(\frac{x^{11}}{x^4 \cdot x^6}\\) #12 \\((a^{-4})^2(a^5)^4\\)
#13 \\(\frac{a^7}{a} \cdot a^4\\) #14 \\(\frac{7^{-3} \cdot 7^8}{7^4 \cdot 7^{-5}}\\) #15 \\((x^3)^4(x^{-3})^{-2}\\) #16 \\((4^3)(4^{-3})\\)
#17 \\((9^2)^5(9^2)(9^4)\\) #18 \\((5^2)^0(5^{-13})(5^7)\\) #19 \\(\frac{17^7}{17^5} \cdot (17^{-5})^2\\) #20 \\(\frac{x \cdot x^5}{x^3 \cdot x^6}\\)

Explanation:

Let's solve these exponent problems one by one using the properties of exponents: \( \frac{a^m}{a^n} = a^{m - n} \), \( a^m \cdot a^n = a^{m + n} \), \( (a^m)^n = a^{m \cdot n} \), and \( a^0 = 1 \) (for \( a
eq 0 \)).

#1 \( \frac{8^4}{8^7} \)

Step 1: Apply the quotient rule for exponents.

The quotient rule states that \( \frac{a^m}{a^n} = a^{m - n} \). Here, \( a = 8 \), \( m = 4 \), and \( n = 7 \).
\( \frac{8^4}{8^7} = 8^{4 - 7} \)

Step 2: Simplify the exponent.

\( 4 - 7 = -3 \), so \( 8^{-3} \). We can also write this as \( \frac{1}{8^3} \) (since \( a^{-n} = \frac{1}{a^n} \)).
\( 8^3 = 512 \), so \( \frac{1}{512} \).

Answer:

\( 8^{-3} \) or \( \frac{1}{512} \)

#2 \( (5^n)(5^3) \)

Step 1: Apply the product rule for exponents.

The product rule states that \( a^m \cdot a^n = a^{m + n} \). Here, \( a = 5 \), \( m = n \), and \( n = 3 \).
\( (5^n)(5^3) = 5^{n + 3} \)