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.