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The image contains a series of math problems related to powers and exponents. I'll guide you through solving some of the problems shown.

### Let's solve a few selected problems:

1. **Problem 1207: Skriv som en potens (Write as a power)**
   - a) \(3^4 \cdot 3^5\)
   - b) \(4^2 \cdot 4^3 \cdot 4^4\)

   **Solution**:
   - a) \(3^4 \cdot 3^5 = 3^{4+5} = 3^9\)
   - b) \(4^2 \cdot 4^3 \cdot 4^4 = 4^{2+3+4} = 4^9\)

2. **Problem 1208: Bestäm exponenten \(m\) så att likheten gäller (Determine the exponent \(m\) so that the equation holds)**
   - a) \(4^m = 4^5 \cdot 4^2\)
   - b) \(10^m = 10^6 \cdot 10^{-4}\)

   **Solution**:
   - a) \(4^m = 4^{5+2} = 4^7\), so \(m = 7\)
   - b) \(10^m = 10^{6 + (-4)} = 10^2\), so \(m = 2\)

3. **Problem 1209: Beräkna utan digitalt hjälpmedel (Calculate without digital tools)**
   - a) \(2^2 \cdot 2^3\)
   - b) \(\frac{5^9}{5^6}\)

   **Solution**:
   - a) \(2^2 \cdot 2^3 = 2^{2+3} = 2^5 = 32\)
   - b) \(\frac{5^9}{5^6} = 5^{9-6} = 5^3 = 125\)

4. **Problem 1214: Skriv som en potens med angiven bas (Write as a power with the given base)**
   - a) \(3^4 \cdot 27\) (Bas 3)
   - b) \(5^7 \cdot 125\) (Bas 5)

   **Solution**:
   - a) \(27 = 3^3\), so \(3^4 \cdot 27 = 3^4 \cdot 3^3 = 3^{4+3} = 3^7\)
   - b) \(125 = 5^3\), so \(5^7 \cdot 125 = 5^7 \cdot 5^3 = 5^{7+3} = 5^{10}\)

5. **Problem 1215: Följande uttryck är en produkt av två potenser. Skriv om uttrycken som en potens av en produkt (The following expressions are a product of two powers. Rewrite the expressions as a power of a product)**
   - a) \(2^3 \cdot x^3\)
   - b) \(3^2 \cdot y^2\)

   **Solution**:
   - a) \(2^3 \cdot x^3 = (2 \cdot x)^3\)
   - b) \(3^2 \cdot y^2 = (3 \cdot y)^2\)

Would you like further details on these solutions or help with any other problems on this page?

### Follow-Up Questions:
1. How would you apply the properties of exponents to simplify expressions like \(a^m \cdot a^n\)?
2. What is the result of dividing powers with the same base, such as \(a^m / a^n\)?
3. Can you determine the value of \(m\) in an equation involving powers with the same base, such as \(b^m = b^p \cdot b^q\)?
4. How can you express a power raised to another power, such as \((a^m)^n\), in a simpler form?
5. What are some real-world applications of powers and exponents?

### Tip:
When simplifying expressions with exponents, always remember the fundamental rules: \(a^m \cdot a^n = a^{m+n}\), \(a^m / a^n = a^{m-n}\), and \((a^m)^n = a^{m \cdot n}\). These will guide you through most exponentiation problems.

Solve the problems related to powers and exponents (potens and exponent) shown in the image.

Learn how to solve a variety of exponent and power problems, including multiplication and division of powers, using key exponent rules. This page provides step-by-step solutions and is suitable for students in Grades 7-9.

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