This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
ScanSolve AI Answer
a) Step 1: Rewrite the terms in the numerator and denominator using exponent rules a^m+n = a^m · a^n. 3^n+1 + 9 · 3^n6 · 3^n + 3^n+1 = (3^n · 3^1 + 9 · 3^n)/(6 · 3^n + 3^n · 3^1) Step 2: Factor out the common term 3^n from the numerator and the denominator. (3^n(3 + 9))/(3^n(6 + 3)) Step 3: Simplify the expression by canceling out 3^n and performing the additions. (3 + 9)/(6 + 3) = (12)/(9) Step 4: Reduce the fraction to its simplest form. (12)/(9) = (4)/(3) The simplified expression is (4)/(3). b) Step 1: Rewrite the terms using exponent rules a^m+n = a^m · a^n and a^xy = (a^x)^y. Note that 25^x = (5^2)^x = 5^2x. 5^2x+1 - 25^x3 · 5^2x+1 + 25^x = 5^2x · 5^1 - 5^2x3 · 5^2x · 5^1 + 5^2x Step 2: Factor out the common term 5^2x from the numerator and the denominator. 5^2x(5 - 1)5^2x(3 · 5 + 1) Step 3: Simplify the expression by canceling out 5^2x and performing the arithmetic. (5 - 1)/(15 + 1) = (4)/(16) Step 4: Reduce the fraction to its simplest form. (4)/(16) = (1)/(4) The simplified expression is (1)/(4). That's 2 down. 3 left today — send the next one.