Question

1. write an equivalent expression for ((-3.1)^{(4x)}cdot(-3.1)^{(2x + 7)})
2. use the law of exponents to write an equivalent expression for (5^{(7 - 3x)}).
3. use the law of exponents to write an equivalent expression for ((28)^{(5x + 8)}).

Explanation:

Step1: Apply product rule of exponents

For same bases, add exponents: $a^m \cdot a^n = a^{m+n}$
$(-3.1)^{(4x)} \cdot (-3.1)^{(2x+7)} = (-3.1)^{4x + 2x + 7}$

Step2: Combine like terms in exponent

Simplify the exponent expression
$(-3.1)^{(4x+2x+7)} = (-3.1)^{6x+7}$

Step3: Apply power of a sum rule

Split exponent using $a^{m-n} = \frac{a^m}{a^n}$
$5^{(7-3x)} = \frac{5^7}{5^{3x}}$
Or use $a^{m+n}=a^m \cdot a^n$: $5^7 \cdot 5^{-3x}$

Step4: Apply power of a sum rule

Split exponent using $a^{m+n}=a^m \cdot a^n$
$(2.8)^{(6x+8)} = (2.8)^{6x} \cdot (2.8)^8$

Answer:

1. $(-3.1)^{6x+7}$
2. $\frac{5^7}{5^{3x}}$ (or $5^7 \cdot 5^{-3x}$)
3. $(2.8)^{6x} \cdot (2.8)^8$