Solve for n: 5 ^ 7 * 5 ^ (5n - 3) / (5 ^ 9) = 1
Understand the Problem
The question asks to solve for 'n' in the given equation involving exponents. We need to simplify the expression using the rules of exponents and then isolate 'n' to find its value.
Answer
$n = 1$
Answer for screen readers
$n = 1$
Steps to Solve
-
Rewrite the equation Begin by rewriting the equation: $$9^{n} \cdot 3^{2} \cdot (3^{-n}) = 27$$
-
Express all terms with the same base Express all terms with base 3: $9 = 3^2$ and $27 = 3^3$. Substitute these into the equation: $$(3^2)^{n} \cdot 3^{2} \cdot 3^{-n} = 3^3$$
-
Simplify the exponents Using the power of a power rule, $(a^m)^n = a^{mn}$, we have: $$3^{2n} \cdot 3^{2} \cdot 3^{-n} = 3^3$$
-
Combine the exponents on the left side Using the rule $a^m \cdot a^n = a^{m+n}$, combine the exponents on the left side: $$3^{2n + 2 - n} = 3^3$$ $$3^{n + 2} = 3^3$$
-
Equate the exponents Since the bases are equal, we can equate the exponents: $$n + 2 = 3$$
-
Solve for n Subtract 2 from both sides of the equation: $$n = 3 - 2$$ $$n = 1$$
$n = 1$
More Information
The value of $n$ that satisfies the given equation is 1.
Tips
A common mistake is not simplifying the exponents correctly. For instance, forgetting to multiply the exponents when dealing with powers of powers, or incorrectly adding or subtracting exponents when multiplying terms with the same base. Another common mistake is skipping the step of converting all terms to the same base which makes the simplification hard to see.
AI-generated content may contain errors. Please verify critical information
