Exponent Rules Review Worksheet with examples and problems for simplifying expressions using exponent rules.

It looks like you've uploaded a worksheet titled "Exponent Rules Review Worksheet", which contains problems related to simplifying expressions using exponent rules. Since I can't view images directly, I’ll solve and explain the problems based on the text you provided.

Let’s go through each problem step by step, applying the laws of exponents:

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🔷 Rules Recap:

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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Now let's solve each problem:

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1) $ a^4 \cdot a^5 $

Use Product Rule:
$ a^4 \cdot a^5 = a^{4+5} = a^9 $

✔ Answer: $ a^9 $

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2) $ (x^3)(x^2) $

Same rule:
$ x^3 \cdot x^2 = x^{3+2} = x^5 $

✔ Answer: $ x^5 $

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3) $ (m^3)^2 $

Use Power of a Power:
$ (m^3)^2 = m^{3 \cdot 2} = m^6 $

✔ Answer: $ m^6 $

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4) $ a^5 \cdot a^3 \cdot a $

Note: $ a = a^1 $, so:
$ a^5 \cdot a^3 \cdot a^1 = a^{5+3+1} = a^9 $

✔ Answer: $ a^9 $

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5) $ (3x^3)(2x^2)(x^3) $

Multiply coefficients and add exponents:
- Coefficients: $ 3 \cdot 2 \cdot 1 = 6 $
- Variables: $ x^3 \cdot x^2 \cdot x^3 = x^{3+2+3} = x^8 $

✔ Answer: $ 6x^8 $

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6) $ (2x^3)^2 $

Use Power of a Product:
$ (2x^3)^2 = 2^2 \cdot (x^3)^2 = 4 \cdot x^{6} = 4x^6 $

✔ Answer: $ 4x^6 $

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7) $ (3x^3)^2 $

Same as above:
$ (3x^3)^2 = 3^2 \cdot (x^3)^2 = 9x^6 $

✔ Answer: $ 9x^6 $

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8) $ (mn)^2 $

Power of a product:
$ (mn)^2 = m^2 n^2 $

✔ Answer: $ m^2 n^2 $

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9) $ (xy)^3 $

$ (xy)^3 = x^3 y^3 $

✔ Answer: $ x^3 y^3 $

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10) $ (7xy)^2 $

$ (7xy)^2 = 7^2 \cdot x^2 \cdot y^2 = 49x^2 y^2 $

✔ Answer: $ 49x^2 y^2 $

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11) $ \frac{x^6}{x^2} $

Use Quotient Rule:
$ \frac{x^6}{x^2} = x^{6-2} = x^4 $

✔ Answer: $ x^4 $

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12) $ \frac{10x^7}{5x^3} $

Divide coefficients: $ \frac{10}{5} = 2 $
Divide variables: $ \frac{x^7}{x^3} = x^{7-3} = x^4 $

So: $ 2x^4 $

✔ Answer: $ 2x^4 $

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13) $ \frac{3x^3}{3x^2} $

Coefficients: $ \frac{3}{3} = 1 $
Variables: $ \frac{x^3}{x^2} = x^{3-2} = x $

So: $ x $

✔ Answer: $ x $

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14) $ \frac{abc^3}{a^2bc^2} $

Break it down:
- $ \frac{a}{a^2} = a^{1-2} = a^{-1} $
- $ \frac{b}{b} = b^0 = 1 $
- $ \frac{c^3}{c^2} = c^{3-2} = c $

So: $ a^{-1} c = \frac{c}{a} $

✔ Answer: $ \frac{c}{a} $

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15) $ \frac{22x^3y^2}{2x^2y} $

Coefficients: $ \frac{22}{2} = 11 $
$ x^3 / x^2 = x^{3-2} = x $
$ y^2 / y = y^{2-1} = y $

So: $ 11xy $

✔ Answer: $ 11xy $

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16) $ x \cdot x^3 $

$ x^1 \cdot x^3 = x^{1+3} = x^4 $

✔ Answer: $ x^4 $

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17) $ (x^3)^4 $

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

✔ Answer: $ x^{12} $

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18) $ (-2x^2)^3 $

$ (-2)^3 = -8 $, and $ (x^2)^3 = x^6 $

So: $ -8x^6 $

✔ Answer: $ -8x^6 $

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19) $ 2x^2 \cdot 7x^2 $

Coefficients: $ 2 \cdot 7 = 14 $
Variables: $ x^2 \cdot x^2 = x^{4} $

So: $ 14x^4 $

✔ Answer: $ 14x^4 $

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20) $ 7^3 $

$ 7^3 = 7 \cdot 7 \cdot 7 = 343 $

✔ Answer: $ 343 $

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21) $ k^4 $

This is already simplified.

✔ Answer: $ k^4 $

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22) $ x^0 $

Any non-zero number to the 0 power is 1.

✔ Answer: $ 1 $

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23) $ (-3)^3 $

$ (-3)^3 = (-3)(-3)(-3) = -27 $

✔ Answer: $ -27 $

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24) $ 6x^2 \cdot 3x^3 $

Coefficients: $ 6 \cdot 3 = 18 $
Variables: $ x^2 \cdot x^3 = x^5 $

So: $ 18x^5 $

✔ Answer: $ 18x^5 $

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25) $ (x - 2)(x - 2x) $

Wait — this is not an exponent problem; it's algebraic multiplication.

But let's simplify:
First, simplify $ x - 2x = -x $

So: $ (x - 2)(-x) = -x(x - 2) = -x^2 + 2x $

✔ Answer: $ -x^2 + 2x $

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26) $ \frac{2x^2}{3x^4} $

Simplify:
- $ \frac{2}{3} $
- $ \frac{x^2}{x^4} = x^{2-4} = x^{-2} = \frac{1}{x^2} $

So: $ \frac{2}{3x^2} $

✔ Answer: $ \frac{2}{3x^2} $

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27) $ \frac{25x^2}{x^3} $

$ \frac{25}{1} \cdot \frac{x^2}{x^3} = 25x^{2-3} = 25x^{-1} = \frac{25}{x} $

✔ Answer: $ \frac{25}{x} $

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28) $ 6a^3 \cdot 3a^2 \cdot a^5 $

Coefficients: $ 6 \cdot 3 = 18 $
Variables: $ a^3 \cdot a^2 \cdot a^5 = a^{3+2+5} = a^{10} $

So: $ 18a^{10} $

✔ Answer: $ 18a^{10} $

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29) $ (3m^2)^3 $

$ 3^3 = 27 $, $ (m^2)^3 = m^6 $

So: $ 27m^6 $

✔ Answer: $ 27m^6 $

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30) $ \left( \frac{3a^2}{b^3} \right)^2 $

Apply Power of a Quotient:
- Numerator: $ (3a^2)^2 = 9a^4 $
- Denominator: $ (b^3)^2 = b^6 $

So: $ \frac{9a^4}{b^6} $

✔ Answer: $ \frac{9a^4}{b^6} $

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## ✔ Final Answers Summary:

| Problem | Answer |
|--------|--------|
| 1 | $ a^9 $ |
| 2 | $ x^5 $ |
| 3 | $ m^6 $ |
| 4 | $ a^9 $ |
| 5 | $ 6x^8 $ |
| 6 | $ 4x^6 $ |
| 7 | $ 9x^6 $ |
| 8 | $ m^2 n^2 $ |
| 9 | $ x^3 y^3 $ |
| 10 | $ 49x^2 y^2 $ |
| 11 | $ x^4 $ |
| 12 | $ 2x^4 $ |
| 13 | $ x $ |
| 14 | $ \frac{c}{a} $ |
| 15 | $ 11xy $ |
| 16 | $ x^4 $ |
| 17 | $ x^{12} $ |
| 18 | $ -8x^6 $ |
| 19 | $ 14x^4 $ |
| 20 | $ 343 $ |
| 21 | $ k^4 $ |
| 22 | $ 1 $ |
| 23 | $ -27 $ |
| 24 | $ 18x^5 $ |
| 25 | $ -x^2 + 2x $ |
| 26 | $ \frac{2}{3x^2} $ |
| 27 | $ \frac{25}{x} $ |
| 28 | $ 18a^{10} $ |
| 29 | $ 27m^6 $ |
| 30 | $ \frac{9a^4}{b^6} $ |

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