Let's go through each of the problems in the image and solve them step by step.

Problem 1212

Problem: Antonia simplifies the expression $(6xy)^2$ to $6x^2y^2$. Explain what she did wrong.

Solution: Antonia's mistake is that she didn't square the constant 6. The correct simplification should be: $(6xy)^2 = 6^2 \cdot x^2 \cdot y^2 = 36x^2y^2.$ So, the correct answer is $36x^2y^2$.

Problem 1213

Simplify as much as possible:

(a) $\frac{2x^4}{x^3}$

Simplify by subtracting the exponents: $\frac{2x^4}{x^3} = 2x^{4-3} = 2x.$

(b) $\frac{x^2 \cdot x^{11}}{x^3}$

Combine the exponents in the numerator and then subtract the exponent in the denominator: $\frac{x^2 \cdot x^{11}}{x^3} = \frac{x^{2+11}}{x^3} = x^{13-3} = x^{10}.$

(c) $\frac{a^2 \cdot a^{11}}{a^3 \cdot a^3 \cdot a^2}$

Combine the exponents in the numerator and denominator: $\frac{a^2 \cdot a^{11}}{a^3 \cdot a^3 \cdot a^2} = \frac{a^{2+11}}{a^{3+3+2}} = \frac{a^{13}}{a^{8}} = a^{13-8} = a^5.$

(d) $\frac{(5x)^2}{x}$

Simplify by expanding and then subtract the exponents: $\frac{(5x)^2}{x} = \frac{25x^2}{x} = 25x^{2-1} = 25x.$

Problem 1214

Write as a power with the given base:

(a) $3^4 \cdot 27$ (base 3)

Since $27 = 3^3$, the expression becomes: $3^4 \cdot 3^3 = 3^{4+3} = 3^7.$

(b) $(5^2)^7 \cdot 125$ (base 5)

Since $125 = 5^3$, the expression becomes: $(5^2)^7 \cdot 5^3 = 5^{14} \cdot 5^3 = 5^{14+3} = 5^{17}.$

(c) $\frac{81}{3^2}$ (base 3)

Since $81 = 3^4$, the expression becomes: $\frac{3^4}{3^2} = 3^{4-2} = 3^2.$

(d) $2 \cdot 32 \cdot 4^2$ (base 2)

Since $32 = 2^5$ and $4 = 2^2$, the expression becomes: $2 \cdot 2^5 \cdot (2^2)^2 = 2^1 \cdot 2^5 \cdot 2^4 = 2^{1+5+4} = 2^{10}.$

Problem 1215

Write the expressions as a single power of a product:

(a) $2^3 \cdot x^3$

Combine the powers: $2^3 \cdot x^3 = (2x)^3.$

(b) $3^2 \cdot y^2$

Combine the powers: $3^2 \cdot y^2 = (3y)^2.$

(c) $a^3 \cdot b^3$

Combine the powers: $a^3 \cdot b^3 = (ab)^3.$

Would you like more details on any of these solutions, or do you have any questions?

Here are some related questions to further your understanding:

1. How can you verify the simplification of exponents when dividing powers with the same base?
2. What is the rule for raising a power to another power in exponentiation?
3. How do you simplify expressions involving products and quotients of exponents?
4. Can you convert expressions to a single power when the bases are different?
5. How do you approach simplifying more complex expressions involving multiple exponent rules?

Tip: When simplifying expressions with exponents, remember to apply the rules systematically: combine like terms, add or subtract exponents where appropriate, and be careful with the bases!