The image you've uploaded is a
math worksheet titled *"Rewriting Using the Laws of Exponents"*, along with an
answer key on the right side. The task involves applying the
laws of exponents to simplify or rewrite expressions.
Let’s go through each problem and explain how the laws of exponents are used to get the correct answer.
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🔷 Laws of Exponents Used:
1.
Zero Exponent Rule:
$ a^0 = 1 $ (for $ a \neq 0 $)
2.
Negative Exponent Rule:
$ a^{-n} = \frac{1}{a^n} $
3.
Product of Powers Rule:
$ a^m \times a^n = a^{m+n} $
4.
Power of a Power Rule:
$ (a^m)^n = a^{m \times n} $
5.
Power of a Product Rule:
$ (ab)^n = a^n \times b^n $
6.
Quotient of Powers Rule:
$ \frac{a^m}{a^n} = a^{m-n} $
7.
Any number to the first power:
$ a^1 = a $
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Now, let's solve each problem
step-by-step:
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1) $ 3^0 = \boxed{1} $
✔ Rule: Zero exponent → any nonzero number to the power of 0 is 1.
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2) $ 3^3 \times 3^{-5} = \boxed{3^{-2}} $
✔ Rule: Product of powers: $ 3^{3 + (-5)} = 3^{-2} $
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3) $ \left(\frac{1}{7}\right)^3 = \boxed{\frac{1}{7^3}} $
✔ Rule: Power of a quotient: $ \left(\frac{a}{b}\right)^n = \frac{a^n}{b^n} $
So $ \left(\frac{1}{7}\right)^3 = \frac{1^3}{7^3} = \frac{1}{7^3} $
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4) $ (7 \times 2)^2 = \boxed{7^2 \times 2^2} $
✔ Rule: Power of a product: $ (ab)^n = a^n b^n $
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5) $ (8 \times 3)^9 = \boxed{8^9 \times 3^9} $
✔ Same as above: $ (ab)^9 = a^9 b^9 $
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6) $ \left(\frac{1}{4}\right)^8 = \boxed{\frac{1}{4^8}} $
✔ $ \left(\frac{1}{4}\right)^8 = \frac{1^8}{4^8} = \frac{1}{4^8} $
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7) $ 8^0 = \boxed{1} $
✔ Zero exponent rule again.
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8) $ 7^8 \times 7^5 = \boxed{7^{13}} $
✔ Product of powers: $ 7^{8+5} = 7^{13} $
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9) $ 5^3 \times 5^2 = \boxed{5^5} $
✔ $ 5^{3+2} = 5^5 $
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10) $ 4^{-8} = \boxed{\frac{1}{4^8}} $
✔ Negative exponent rule: $ a^{-n} = \frac{1}{a^n} $
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11) $ 7^1 = \boxed{7} $
✔ Any number to the power of 1 is itself.
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12) $ (8 \times 6)^9 = \boxed{8^9 \times 6^9} $
✔ Power of a product: $ (ab)^9 = a^9 b^9 $
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13) $ 4^8 \times 4^{-7} = \boxed{4^1} $
✔ $ 4^{8 + (-7)} = 4^1 = 4 $
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14) $ 4^4 \times 4^3 = \boxed{4^7} $
✔ $ 4^{4+3} = 4^7 $
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15) $ (8^4)^3 = \boxed{8^{12}} $
✔ Power of a power: $ (a^m)^n = a^{m \cdot n} $ → $ 8^{4 \cdot 3} = 8^{12} $
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16) $ 6^3 \times 6^{-9} = \boxed{6^{-6}} $
✔ $ 6^{3 + (-9)} = 6^{-6} $
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17) $ 3^0 = \boxed{1} $
✔ Zero exponent rule
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18) $ 6^1 = \boxed{6} $
✔ Any number to the first power is itself
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19) $ 8^{-5} = \boxed{\frac{1}{8^5}} $
✔ Negative exponent rule: $ a^{-n} = \frac{1}{a^n} $
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20) $ (5^3)^5 = \boxed{5^{15}} $
✔ Power of a power: $ (a^m)^n = a^{m \cdot n} $ → $ 5^{3 \cdot 5} = 5^{15} $
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✔ Final Notes:
- All answers in the
Answer Key column match the correct application of exponent rules.
- This worksheet reinforces understanding of
basic exponent laws, especially:
- Multiplying like bases
- Negative exponents
- Powers of products and powers
- Zero and first powers
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📌 Summary of Key Concepts:
| Expression | Rule Applied | Result |
|-----------|--------------|--------|
| $ a^0 $ | Zero exponent | 1 |
| $ a^{-n} $ | Negative exponent | $ \frac{1}{a^n} $ |
| $ a^m \cdot a^n $ | Product of powers | $ a^{m+n} $ |
| $ (a^m)^n $ | Power of a power | $ a^{m \cdot n} $ |
| $ (ab)^n $ | Power of a product | $ a^n b^n $ |
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✔ All problems are correctly solved using the laws of exponents.
The answer key provided is accurate.
Let me know if you'd like this turned into a printable study guide or want practice problems!
Parent Tip: Review the logic above to help your child master the concept of exponents rules worksheet.
