https://file.aiquickdraw.com/mathbot/file/1725902239247iosfz83s.jpg

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!

Multiple problems on simplifying algebraic expressions and exponents in Swedish from the given image.

This webpage provides step-by-step solutions to simplifying algebraic expressions with exponents, using laws of exponents for multiplication, division, and powers of products. Perfect for Grades 9-11 math students looking to strengthen their understanding.

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