The task involves simplifying expressions using the laws of exponents. Let's go through each problem step by step and explain the solution.
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Problem 1: Simplify \( 4x^3 \cdot 2x^3 \)
Solution:
1. Multiply the coefficients: \( 4 \cdot 2 = 8 \).
2. Add the exponents of \( x \) (since the bases are the same): \( x^3 \cdot x^3 = x^{3+3} = x^6 \).
3. Combine the results: \( 8x^6 \).
Answer: \( 8x^6 \)
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Problem 2: Simplify \( x^5 \cdot x^3 \)
Solution:
1. Add the exponents of \( x \) (since the bases are the same): \( x^5 \cdot x^3 = x^{5+3} = x^8 \).
Answer: \( x^8 \)
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Problem 3: Simplify \( 2x^3 \cdot 2x^2 \)
Solution:
1. Multiply the coefficients: \( 2 \cdot 2 = 4 \).
2. Add the exponents of \( x \) (since the bases are the same): \( x^3 \cdot x^2 = x^{3+2} = x^5 \).
3. Combine the results: \( 4x^5 \).
Answer: \( 4x^5 \)
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Problem 4: Simplify \( \frac{6^5}{6^3} \)
Solution:
1. Subtract the exponents of the base \( 6 \) (since the bases are the same and it's a division): \( \frac{6^5}{6^3} = 6^{5-3} = 6^2 \).
2. Calculate \( 6^2 \): \( 6^2 = 36 \).
Answer: \( 36 \)
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Problem 5: Simplify \( \frac{x^4}{x^7} \)
Solution:
1. Subtract the exponents of \( x \) (since the bases are the same and it's a division): \( \frac{x^4}{x^7} = x^{4-7} = x^{-3} \).
2. Rewrite \( x^{-3} \) as \( \frac{1}{x^3} \).
Answer: \( \frac{1}{x^3} \)
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Problem 6: Simplify \( 8^0 \)
Solution:
1. Any non-zero number raised to the power of 0 is 1: \( 8^0 = 1 \).
Answer: \( 1 \)
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Problem 7: Simplify \( \frac{6x^7}{2x^4} \)
Solution:
1. Divide the coefficients: \( \frac{6}{2} = 3 \).
2. Subtract the exponents of \( x \) (since the bases are the same and it's a division): \( \frac{x^7}{x^4} = x^{7-4} = x^3 \).
3. Combine the results: \( 3x^3 \).
Answer: \( 3x^3 \)
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Problem 8: Simplify \( \frac{x^2y^5}{xy^4} \)
Solution:
1. Divide the coefficients of \( x \): \( \frac{x^2}{x} = x^{2-1} = x^1 = x \).
2. Divide the coefficients of \( y \): \( \frac{y^5}{y^4} = y^{5-4} = y^1 = y \).
3. Combine the results: \( xy \).
Answer: \( xy \)
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Final Answers:
1. \( 8x^6 \)
2. \( x^8 \)
3. \( 4x^5 \)
4. \( 36 \)
5. \( \frac{1}{x^3} \)
6. \( 1 \)
7. \( 3x^3 \)
8. \( xy \)
\boxed{8x^6, x^8, 4x^5, 36, \frac{1}{x^3}, 1, 3x^3, xy}
Parent Tip: Review the logic above to help your child master the concept of rules of exponents worksheet pdf.
