Question
Use the grouping symbols to interpret the following equation: x8=3(x+4)2 . Which expression in the equation represents a product?
x8
2
3(x+4)
3(x+4)2
x8
2
3(x+4)
3(x+4)2
Answers
GPT-4o mini
To determine which expression in the equation \( x^8 = 3(x+4)^2 \) represents a product, we need to identify the parts of the equation that involve multiplication.
Let's break down the given expression \( 3(x+4)^2 \):
1. **\( 3 \)**: This is a constant.
2. **\( (x+4) \)**: This is a binomial expression.
3. **\( (x+4)^2 \)**: This involves raising the binomial to the power of 2, which can be rewritten as \( (x+4) \cdot (x+4) \) — this shows that it is indeed a product of two identical terms.
4. **The complete expression \( 3(x+4)^2 \)** represents a product of the number 3 and the square of the binomial \( (x+4) \).
Now comparing the options:
- **\( x^8 \)**: This is not a product; it's a single term raised to a power.
- **\( 2 \)**: This is a constant and does not represent any product in this context.
- **\( 3(x+4) \)**: This involves a multiplication of 3 and the expression \( (x+4) \), which represents a product.
- **\( 3(x+4)^2 \)**: This represents a product of 3 and the square of \( (x+4) \).
Among the choices listed, both \( 3(x+4) \) and \( 3(x+4)^2 \) represent products. However, the expression that most clearly represents a product involving both multiplication and addition is:
**Answer: \( 3(x+4)^2 \)** (because it represents the multiplication of 3 and the square of the binomial).
Let's break down the given expression \( 3(x+4)^2 \):
1. **\( 3 \)**: This is a constant.
2. **\( (x+4) \)**: This is a binomial expression.
3. **\( (x+4)^2 \)**: This involves raising the binomial to the power of 2, which can be rewritten as \( (x+4) \cdot (x+4) \) — this shows that it is indeed a product of two identical terms.
4. **The complete expression \( 3(x+4)^2 \)** represents a product of the number 3 and the square of the binomial \( (x+4) \).
Now comparing the options:
- **\( x^8 \)**: This is not a product; it's a single term raised to a power.
- **\( 2 \)**: This is a constant and does not represent any product in this context.
- **\( 3(x+4) \)**: This involves a multiplication of 3 and the expression \( (x+4) \), which represents a product.
- **\( 3(x+4)^2 \)**: This represents a product of 3 and the square of \( (x+4) \).
Among the choices listed, both \( 3(x+4) \) and \( 3(x+4)^2 \) represent products. However, the expression that most clearly represents a product involving both multiplication and addition is:
**Answer: \( 3(x+4)^2 \)** (because it represents the multiplication of 3 and the square of the binomial).