Apply the Properties of Integer Exponents to generate equivalent expressions to (43 ⋅ 7−2)−579 ⋅ 4−13 with only positive exponents, then solve the expression. Your answer will be a simplified fraction with no remaining exponents.

Question

(43 ⋅ 7−2)−579 ⋅ 4−13=

Answer 1

To generate equivalent expressions with only positive exponents, we can use the following properties of integer exponents:

1. Product of Powers Property: $(a^m)(a^n) = a^{m + n}$
2. Quotient of Powers Property: $\dfrac{a^m}{a^n} = a^{m - n}$
3. Power of a Power Property: $(a^m)^n = a^{m \cdot n}$
4. Power of a Product Property: $(ab)^n = a^n \cdot b^n$

Using these properties, we can simplify the given expression step by step:

$$(43 \cdot 7^{-2})^{-579} \cdot 4^{-13}$$
$$=\left(\dfrac{43}{7^2}\right)^{-579} \cdot \left(\dfrac{1}{4^{13}}\right)$$
$$=\left(\dfrac{43}{7^2}\right)^{-579} \cdot \left(\dfrac{1}{2^{26}}\right)$$
$$=\left(\dfrac{43}{49}\right)^{-579} \cdot \left(\dfrac{1}{2^{26}}\right)$$

Now, let's simplify the expression by using negative exponent rule which states that $a^{-n} = \dfrac{1}{a^n}$:

$$\left(\dfrac{49}{43}\right)^{579} \cdot 2^{26}$$

Therefore, the simplified expression is $\left(\dfrac{49}{43}\right)^{579} \cdot 2^{26}$.