Let's solve each problem step by step:
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Problem 1: Using a Factor Tree, find the prime factorization of 54.
#### Solution:
To find the prime factorization of 54 using a factor tree:
1. Start with 54.
2. Break it down into two factors: \( 54 = 6 \times 9 \).
3. Further break down 6 and 9 into their prime factors:
- \( 6 = 2 \times 3 \)
- \( 9 = 3 \times 3 \)
4. Combine all the prime factors: \( 54 = 2 \times 3 \times 3 \times 3 \).
5. Write this in exponential form: \( 54 = 2^1 \times 3^3 \).
However, the options provided are slightly different. Let's match them:
- Option A: \( 2^3 \times 3 \) (Incorrect, as \( 2^3 = 8 \)).
- Option B: \( 3^2 \times 6 \) (Incorrect, as 6 is not a prime number).
- Option C: \( 9 \times 6 \) (Incorrect, as neither 9 nor 6 is a prime number).
- Option D: \( 3^3 \times 2 \) (Correct, as \( 3^3 = 27 \) and \( 27 \times 2 = 54 \)).
Answer: D
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Problem 2: What is the solution of \( 9 + k = 18 \)?
#### Solution:
To solve for \( k \):
1. Start with the equation: \( 9 + k = 18 \).
2. Subtract 9 from both sides: \( k = 18 - 9 \).
3. Simplify: \( k = 9 \).
Answer: C
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Problem 3: Evaluate the expression \( (4^3 - 4) + 12 \div 3 \).
#### Solution:
Follow the order of operations (PEMDAS/BODMAS):
1. Calculate \( 4^3 \): \( 4^3 = 64 \).
2. Subtract 4: \( 64 - 4 = 60 \).
3. Divide 12 by 3: \( 12 \div 3 = 4 \).
4. Add the results: \( 60 + 4 = 64 \).
Answer: D
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Problem 4: What is the solution of \( 3w = 30 \)?
#### Solution:
To solve for \( w \):
1. Start with the equation: \( 3w = 30 \).
2. Divide both sides by 3: \( w = \frac{30}{3} \).
3. Simplify: \( w = 10 \).
Answer: A
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Problem 5: What is the solution of \( \frac{d}{5} = 10 \)?
#### Solution:
To solve for \( d \):
1. Start with the equation: \( \frac{d}{5} = 10 \).
2. Multiply both sides by 5: \( d = 10 \times 5 \).
3. Simplify: \( d = 50 \).
Answer: C
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Problem 6: Which of the following are expressions? (Select all that apply.)
#### Solution:
An
expression is a mathematical phrase that combines numbers, variables, and operators but does not contain an equals sign (=).
- Option A: "twelve is three more than nine" → This is a statement, not an expression (contains "is").
- Option B: \( 3x + 4 \) → This is an expression (contains variables and operators).
- Option C: "the difference of \( x \) and 8" → This is a description, not an expression (contains words like "difference").
- Option D: \( 2y \) → This is an expression (contains a variable and a coefficient).
Answer: B, D
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Problem 7: What is the solution of \( 13 = t + 7 \)?
#### Solution:
To solve for \( t \):
1. Start with the equation: \( 13 = t + 7 \).
2. Subtract 7 from both sides: \( t = 13 - 7 \).
3. Simplify: \( t = 6 \).
Answer: 6
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Final Answers:
1.
D
2.
C
3.
D
4.
A
5.
C
6.
B, D
7.
6
Boxed Final Answer:
\[
\boxed{D, C, D, A, C, B, D, 6}
\]
Parent Tip: Review the logic above to help your child master the concept of expressions and equations 6th grade worksheet.
