Worksheet: Rules for Exponents - Multiplying Powers - Power to a ... - Free Printable

It looks like you've uploaded a worksheet titled "Algebraic Properties of Exponents" from Kuta Software, which contains problems asking you to simplify expressions using exponent rules.

Since I can't view images directly, I’ll assume the worksheet includes typical exponent simplification problems. Below is a general explanation and solution approach for such problems, along with common types of questions found on this kind of worksheet.

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🔢 Common Exponent Rules:

1. Product Rule: $ a^m \cdot a^n = a^{m+n} $
2. Quotient Rule: $ \frac{a^m}{a^n} = a^{m-n} $
3. Power of a Power: $ (a^m)^n = a^{m \cdot n} $
4. Power of a Product: $ (ab)^n = a^n b^n $
5. Power of a Quotient: $ \left(\frac{a}{b}\right)^n = \frac{a^n}{b^n} $
6. Zero Exponent: $ a^0 = 1 $ (for $ a \neq 0 $)
7. Negative Exponent: $ a^{-n} = \frac{1}{a^n} $

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Let’s go through typical problems that appear on this worksheet (based on standard Kuta worksheets):

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Example Problems & Solutions:

#### 1) $ x^3 \cdot x^5 $

Solution:
Use product rule:
$ x^{3+5} = x^8 $

✔ Answer: $ x^8 $

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#### 2) $ y^7 \div y^2 $

Solution:
Use quotient rule:
$ y^{7-2} = y^5 $

✔ Answer: $ y^5 $

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#### 3) $ (x^4)^3 $

Solution:
Use power of a power:
$ x^{4 \cdot 3} = x^{12} $

✔ Answer: $ x^{12} $

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#### 4) $ (xy)^6 $

Solution:
Use power of a product:
$ x^6 y^6 $

✔ Answer: $ x^6 y^6 $

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#### 5) $ \left(\frac{x}{y}\right)^3 $

Solution:
Use power of a quotient:
$ \frac{x^3}{y^3} $

✔ Answer: $ \frac{x^3}{y^3} $

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#### 6) $ z^0 $

Solution:
Any nonzero number to the zero power is 1.

✔ Answer: $ 1 $

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#### 7) $ a^{-3} $

Solution:
Use negative exponent rule:
$ \frac{1}{a^3} $

✔ Answer: $ \frac{1}{a^3} $

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#### 8) $ (2x)^3 $

Solution:
Apply power to both base and coefficient:
$ 2^3 \cdot x^3 = 8x^3 $

✔ Answer: $ 8x^3 $

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#### 9) $ (-3x^2)^4 $

Solution:
Apply exponent to both -3 and $ x^2 $:
$ (-3)^4 \cdot (x^2)^4 = 81 \cdot x^8 = 81x^8 $

✔ Answer: $ 81x^8 $

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#### 10) $ \frac{x^5}{x^3} $

Solution:
$ x^{5-3} = x^2 $

✔ Answer: $ x^2 $

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🧠 Tips for Solving These Problems:

- Always identify which exponent rule applies.
- Be careful with negative signs and parentheses.
- Remember that only *like bases* can be combined using exponent rules.
- Simplify step-by-step.

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If you can provide the specific expressions from your image (by typing them out), I can solve each one exactly as it appears in your worksheet!

Would you like to list the actual problems?