Simplifying Expressions with Exponents: (m^5n^3)^7 times m^2n
In mathematics, simplifying expressions involving exponents is a fundamental skill. Let's explore how to simplify the expression (m^5n^3)^7 times m^2n.
Understanding the Rules of Exponents
Before we tackle the problem, let's review some key exponent rules:
 Product of Powers: When multiplying powers with the same base, you add the exponents: x^m * x^n = x^(m+n)
 Power of a Power: When raising a power to another power, you multiply the exponents: (x^m)^n = x^(m*n)
Simplifying the Expression

Apply the Power of a Power Rule: First, we address the outer exponent in (m^5n^3)^7. Applying the rule, we get: (m^5n^3)^7 = m^(57)n^(37) = m^35n^21

Apply the Product of Powers Rule: Now we have m^35n^21 * m^2n. Since the bases are the same (m and n), we add the exponents: m^35n^21 * m^2n = m^(35+2)n^(21+1) = m^37n^22
Final Result
Therefore, the simplified expression of (m^5n^3)^7 times m^2n is m^37n^22.