Here are the solutions to the simplification problems. d) Simplify $M^0$ Step 1: Apply the zero exponent rule. Any non-zero number raised to the power of zero is equal to 1. Step 2: Apply the rule to the expression. $$M^0 = 1 \quad (\text{assuming } M \neq 0)$$ The simplified expression is $\boxed{1}$. The statement "$M^0 > 1$" is incorrect, as $M^0$ is equal to $1$. e) Simplify $\frac{2^5 \times 2^3}{\sqrt{(1+2)}}$ Step 1: Simplify the numerator using the product rule for exponents ($a^m \times a^n = a^{m+n}$). $$2^5 \times 2^3 = 2^{5+3} = 2^8$$ Step 2: Calculate the value of $2^8$. $$2^8 = 256$$ Step 3: Simplify the denominator. $$\sqrt{(1+2)} = \sqrt{3}$$ Step 4: Combine the simplified numerator and denominator. $$\frac{256}{\sqrt{3}}$$ Step 5: Rationalize the denominator by multiplying the numerator and denominator by $\sqrt{3}$. $$\frac{256}{\sqrt{3}} \times \frac{\sqrt{3}}{\sqrt{3}} = \frac{256\sqrt{3}}{3}$$ The simplified expression is $\boxed{\frac{256\sqrt{3}}{3}}$. g) Simplify $(8-3)^3$ Step 1: Perform the subtraction inside the parentheses. $$8-3 = 5$$ Step 2: Raise the result to the power of 3. $$5^3 = 5 \times 5 \times 5$$ Step 3: Calculate the product. $$5 \times 5 \times 5 = 25 \times 5 = 125$$ The simplified expression is $\boxed{125}$. The statement "> 125" is incorrect, as the result is exactly $125$. 3 done, 2 left today. You're making progress.