Math worksheet on laws of exponents with problems for simplifying expressions using various exponent rules.

Laws of Exponents Worksheet Solution

The worksheet involves simplifying expressions using various laws of exponents. Let's solve each problem step by step.

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#### Simplify using product of power rule

The product of powers rule states:
\[ a^m \cdot a^n = a^{m+n} \]

1. Simplify \( y^{40} \cdot y^{20} \)
Using the product of powers rule:
\[ y^{40} \cdot y^{20} = y^{40+20} = y^{60} \]
Answer: \( y^{60} \)

2. Simplify \( 3x^2 \cdot 2y^4 \)
Combine the coefficients and keep the variables separate:
\[ 3x^2 \cdot 2y^4 = (3 \cdot 2) \cdot (x^2) \cdot (y^4) = 6x^2y^4 \]
Answer: \( 6x^2y^4 \)

3. Simplify \( (3abc)(2a^2b) \)
Combine the coefficients and use the product of powers rule for the variables:
\[ (3abc)(2a^2b) = (3 \cdot 2) \cdot (a \cdot a^2) \cdot (b \cdot b) \cdot c = 6a^{1+2}b^{1+1}c = 6a^3b^2c \]
Answer: \( 6a^3b^2c \)

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#### Simplify using quotient of powers rule

The quotient of powers rule states:
\[ \frac{a^m}{a^n} = a^{m-n} \]

4. Simplify \( \frac{x^4y^7}{x^3y^3} \)
Apply the quotient of powers rule to each variable:
\[ \frac{x^4y^7}{x^3y^3} = x^{4-3} \cdot y^{7-3} = x^1 \cdot y^4 = xy^4 \]
Answer: \( xy^4 \)

5. Simplify \( \frac{a^{10}b^{20}}{a^2b^{10}} \)
Apply the quotient of powers rule to each variable:
\[ \frac{a^{10}b^{20}}{a^2b^{10}} = a^{10-2} \cdot b^{20-10} = a^8 \cdot b^{10} = a^8b^{10} \]
Answer: \( a^8b^{10} \)

6. Simplify \( \frac{52a^{10}b^2}{5ab^2} \)
Simplify the coefficients and apply the quotient of powers rule to each variable:
\[ \frac{52a^{10}b^2}{5ab^2} = \left( \frac{52}{5} \right) \cdot a^{10-1} \cdot b^{2-2} = \frac{52}{5} \cdot a^9 \cdot b^0 = \frac{52}{5}a^9 \cdot 1 = \frac{52}{5}a^9 \]
Answer: \( \frac{52}{5}a^9 \)

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#### Simplify using negative and zero exponent rule

The zero exponent rule states:
\[ a^0 = 1 \quad \text{(for } a \neq 0\text{)} \]

The negative exponent rule states:
\[ a^{-n} = \frac{1}{a^n} \]

7. Simplify \( (-2)^0 \)
Using the zero exponent rule:
\[ (-2)^0 = 1 \]
Answer: \( 1 \)

8. Simplify \( (-3)^{-4} \)
Using the negative exponent rule:
\[ (-3)^{-4} = \frac{1}{(-3)^4} = \frac{1}{81} \]
Answer: \( \frac{1}{81} \)

9. Simplify \( \frac{1}{4^{-2}} \)
Using the negative exponent rule:
\[ \frac{1}{4^{-2}} = 4^2 = 16 \]
Answer: \( 16 \)

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#### Simplify using power rule

The power rule states:
\[ (a^m)^n = a^{m \cdot n} \]

10. Simplify \( (4x^2)^6 \)
Apply the power rule to both the coefficient and the variable:
\[ (4x^2)^6 = 4^6 \cdot (x^2)^6 = 4^6 \cdot x^{2 \cdot 6} = 4^6 \cdot x^{12} \]
Calculate \( 4^6 \):
\[ 4^6 = (2^2)^6 = 2^{12} = 4096 \]
So,
\[ (4x^2)^6 = 4096x^{12} \]
Answer: \( 4096x^{12} \)

11. Simplify \( (6x^2y^4)^2 \)
Apply the power rule to the coefficient and each variable:
\[ (6x^2y^4)^2 = 6^2 \cdot (x^2)^2 \cdot (y^4)^2 = 36 \cdot x^{2 \cdot 2} \cdot y^{4 \cdot 2} = 36x^4y^8 \]
Answer: \( 36x^4y^8 \)

12. Simplify \( (x^{-4})^{-4} \)
Apply the power rule:
\[ (x^{-4})^{-4} = x^{-4 \cdot -4} = x^{16} \]
Answer: \( x^{16} \)

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Final Answers:

1. \( y^{60} \)
2. \( 6x^2y^4 \)
3. \( 6a^3b^2c \)
4. \( xy^4 \)
5. \( a^8b^{10} \)
6. \( \frac{52}{5}a^9 \)
7. \( 1 \)
8. \( \frac{1}{81} \)
9. \( 16 \)
10. \( 4096x^{12} \)
11. \( 36x^4y^8 \)
12. \( x^{16} \)

Boxed Final Answer:
\[
\boxed{
\begin{array}{ll}
1. & y^{60} \\
2. & 6x^2y^4 \\
3. & 6a^3b^2c \\
4. & xy^4 \\
5. & a^8b^{10} \\
6. & \frac{52}{5}a^9 \\
7. & 1 \\
8. & \frac{1}{81} \\
9. & 16 \\
10. & 4096x^{12} \\
11. & 36x^4y^8 \\
12. & x^{16} \\
\end{array}
}
\]