https://file.aiquickdraw.com/mathbot/file/17302172839630dsalz18.jpg

Let's go through each question from the uploaded image and solve them one by one:

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### 4.1
\[
\left( \frac{5}{13} \right)^0
\]
Anything raised to the power of zero is 1, so:
\[
\left( \frac{5}{13} \right)^0 = 1
\]

---

### 4.2
\[
2^x \cdot 5^x
\]
We can combine the terms since both have the exponent \( x \):
\[
2^x \cdot 5^x = (2 \cdot 5)^x = 10^x
\]

---

### 4.3
\[
\frac{1}{2 + 2^{-1}}
\]
First, simplify \( 2^{-1} = \frac{1}{2} \):
\[
2 + 2^{-1} = 2 + \frac{1}{2} = \frac{4 + 1}{2} = \frac{5}{2}
\]
So, the expression becomes:
\[
\frac{1}{\frac{5}{2}} = \frac{2}{5}
\]

---

### 4.4
\[
\frac{2^{2n-1} \cdot 4^{n+1} \cdot 2}{4^n}
\]
Rewrite \( 4^{n+1} \) and \( 4^n \) in terms of base 2, since \( 4 = 2^2 \):
\[
4^{n+1} = (2^2)^{n+1} = 2^{2(n+1)} = 2^{2n+2}
\]
\[
4^n = (2^2)^n = 2^{2n}
\]
Substitute back into the expression:
\[
\frac{2^{2n-1} \cdot 2^{2n+2} \cdot 2}{2^{2n}}
\]
Combine the exponents in the numerator:
\[
= \frac{2^{(2n-1) + (2n+2) + 1}}{2^{2n}} = \frac{2^{4n+2}}{2^{2n}} = 2^{(4n+2) - 2n} = 2^{2n+2}
\]

---

### 4.5
\[
\frac{(a^3)^{n+1} \cdot (ab^2)^{-2n}}{(ab)^n \cdot b^{-5n}}
\]
Rewrite each part using power rules:
1. \((a^3)^{n+1} = a^{3(n+1)} = a^{3n+3}\)
2. \((ab^2)^{-2n} = a^{-2n} \cdot b^{-4n}\)
3. \((ab)^n = a^n \cdot b^n\)

Substitute and simplify:
\[
= \frac{a^{3n+3} \cdot a^{-2n} \cdot b^{-4n}}{a^n \cdot b^n \cdot b^{-5n}}
\]
Combine \(a\) and \(b\) exponents separately:
\[
= a^{(3n+3 - 2n - n)} \cdot b^{(-4n - n + 5n)} = a^3 \cdot b^0 = a^3
\]

---

### 4.6
\[
\frac{2 \cdot 3^x}{3^{x+2} - 3^x}
\]
Factor \(3^x\) out of the denominator:
\[
= \frac{2 \cdot 3^x}{3^x (3^2 - 1)} = \frac{2 \cdot 3^x}{3^x \cdot 8} = \frac{2}{8} = \frac{1}{4}
\]

---

### 4.7
Solve for \( x \):
\[
2^x = 64
\]
Rewrite \(64\) as a power of \(2\):
\[
64 = 2^6
\]
Thus:
\[
2^x = 2^6 \Rightarrow x = 6
\]

---

### 4.8
Solve for \( x \):
\[
3^x = \frac{1}{81}
\]
Rewrite \( \frac{1}{81} \) as \( 3^{-4} \):
\[
3^x = 3^{-4} \Rightarrow x = -4
\]

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Would you like a deeper explanation for any of these steps?

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#### Additional Practice Questions:
1. Simplify \( \left(\frac{3}{4}\right)^{-2} \).
2. Solve for \( x \) in \( 5^{2x} = 125 \).
3. Rewrite \( (x^2 y^{-3})^2 \) with positive exponents only.
4. Evaluate \( \frac{16^{-1/2}}{4^{-1}} \).
5. Solve \( 7^{x+1} = 343 \) for \( x \).

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#### Tip:
When simplifying expressions with exponents, remember that terms with the same base can be combined by adding or subtracting their exponents based on multiplication or division.

Simplify and solve exponent expressions without calculators, including expressions like (5/13)^0, 2^x * 5^x, and equations such as 2^x = 64 and 3^x = 1/81.

Explore step-by-step solutions for simplifying expressions and solving equations with exponents without a calculator. Topics include the zero exponent rule, product and division of powers, and algebraic simplifications for questions like (5/13)^0 and 2^x = 64. Perfect for high school students in Grades 10-12.

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