Apply the properties of integer exponents to generate an equivalent to (5^-³)⁶ • 5¹⁷/10^-² with only positive exponents, then solve the expression.

Question

(5^-³)⁶ • 5¹⁷/10^-²=□

Answer 1

To generate an equivalent expression with only positive exponents, we can apply the properties of integer exponents.

First, let's simplify the expression (5^-³)⁶ • 5¹⁷/10^-²:

(5^-³)⁶ • 5¹⁷/10^-²

The property of raising a power to a power states that (a^m)^n = a^(m * n). Applying this property to (5^-³)⁶:

(5^-³)⁶ = 5^(-³ * 6) = 5^-18

Now the expression becomes:

5^-18 • 5¹⁷/10^-²

The property of dividing powers with the same base states that a^m / a^n = a^(m - n). Applying this property to 5^-18 / 10^-²:

5^-18 / 10^-² = 5^(-18 - (-2)) = 5^-18 + 2 = 5^-16

Now the expression becomes:

5^-16 • 5¹⁷

The property of multiplying powers with the same base states that a^m * a^n = a^(m + n). Applying this property to 5^-16 • 5¹⁷:

5^-16 • 5¹⁷ = 5^(-16 + 17) = 5^1 = 5

Therefore, the simplified expression is 5.