Simplify 6^2 - 3^2
- A. 6
- B. 9
- C. 27
- D. 3
Correct Answer & Rationale
Correct Answer: C
To simplify \(6^2 - 3^2\), we apply the difference of squares formula, which states \(a^2 - b^2 = (a - b)(a + b)\). Here, \(a = 6\) and \(b = 3\). Thus, we have: \[ 6^2 - 3^2 = (6 - 3)(6 + 3) = 3 \times 9 = 27 \] Option A (6) is incorrect as it miscalculates the expression. Option B (9) mistakenly considers only one of the squared terms. Option D (3) misinterprets the operations involved, leading to an incorrect result. The correct evaluation yields 27, confirming option C as the accurate answer.
To simplify \(6^2 - 3^2\), we apply the difference of squares formula, which states \(a^2 - b^2 = (a - b)(a + b)\). Here, \(a = 6\) and \(b = 3\). Thus, we have: \[ 6^2 - 3^2 = (6 - 3)(6 + 3) = 3 \times 9 = 27 \] Option A (6) is incorrect as it miscalculates the expression. Option B (9) mistakenly considers only one of the squared terms. Option D (3) misinterprets the operations involved, leading to an incorrect result. The correct evaluation yields 27, confirming option C as the accurate answer.
Other Related Questions
What is the value of 2/5 multiplied by 5/4 divided by 4/3
- A. 32/75
- B. 3\8
- C. 6\25
- D. 2\3
Correct Answer & Rationale
Correct Answer: B
To solve \( \frac{2}{5} \times \frac{5}{4} \div \frac{4}{3} \), we first multiply \( \frac{2}{5} \) by \( \frac{5}{4} \). This results in \( \frac{2 \times 5}{5 \times 4} = \frac{10}{20} = \frac{1}{2} \). Next, dividing by \( \frac{4}{3} \) is the same as multiplying by its reciprocal, \( \frac{3}{4} \). Therefore, \( \frac{1}{2} \times \frac{3}{4} = \frac{3}{8} \). Option A, \( \frac{32}{75} \), is incorrect as it does not simplify from the given operations. Option C, \( \frac{6}{25} \), results from miscalculating the division. Option D, \( \frac{2}{3} \), is also incorrect as it doesn't follow from the correct operations.
To solve \( \frac{2}{5} \times \frac{5}{4} \div \frac{4}{3} \), we first multiply \( \frac{2}{5} \) by \( \frac{5}{4} \). This results in \( \frac{2 \times 5}{5 \times 4} = \frac{10}{20} = \frac{1}{2} \). Next, dividing by \( \frac{4}{3} \) is the same as multiplying by its reciprocal, \( \frac{3}{4} \). Therefore, \( \frac{1}{2} \times \frac{3}{4} = \frac{3}{8} \). Option A, \( \frac{32}{75} \), is incorrect as it does not simplify from the given operations. Option C, \( \frac{6}{25} \), results from miscalculating the division. Option D, \( \frac{2}{3} \), is also incorrect as it doesn't follow from the correct operations.
What is the value of f(-3) for f(x) = 2x^2 + x + 1
Correct Answer & Rationale
Correct Answer: -20
To find \( f(-3) \) for the function \( f(x) = 2x^2 + x + 1 \), substitute \(-3\) for \(x\): \[ f(-3) = 2(-3)^2 + (-3) + 1 = 2(9) - 3 + 1 = 18 - 3 + 1 = 16. \] The correct answer is -20, which is incorrect based on the calculation. Examining the other options: - If an option were 16, it would be correct as shown in the calculation. - Any other number, like -10 or 0, would arise from miscalculations or incorrect substitutions, thus not representing the function's value at \(-3\). The accurate evaluation confirms that \( f(-3) = 16 \).
To find \( f(-3) \) for the function \( f(x) = 2x^2 + x + 1 \), substitute \(-3\) for \(x\): \[ f(-3) = 2(-3)^2 + (-3) + 1 = 2(9) - 3 + 1 = 18 - 3 + 1 = 16. \] The correct answer is -20, which is incorrect based on the calculation. Examining the other options: - If an option were 16, it would be correct as shown in the calculation. - Any other number, like -10 or 0, would arise from miscalculations or incorrect substitutions, thus not representing the function's value at \(-3\). The accurate evaluation confirms that \( f(-3) = 16 \).
A shipping box for a refrigerator is shaped like a rectangular prism. The box has a depth of 34.25 inches (in.), a height of 69.37 in., and a width of 32.62 in. To the nearest hundredth cubic inch, what is the volume of the shipping bax?
- A. 2,262.85
- B. 77,502.59
- C. 136.24
- D. 25,834.20
Correct Answer & Rationale
Correct Answer: B
To determine the volume of a rectangular prism, the formula \( V = \text{length} \times \text{width} \times \text{height} \) is applied. Given the dimensions—depth (length) of 34.25 in., width of 32.62 in., and height of 69.37 in.—the calculation yields a volume of approximately 77,502.59 cubic inches. Option A (2,262.85) is far too small, indicating a miscalculation. Option C (136.24) is implausibly low, likely resulting from using incorrect units or dimensions. Option D (25,834.20) is also incorrect, as it does not reflect the correct multiplication of the given dimensions. Thus, only option B accurately represents the calculated volume.
To determine the volume of a rectangular prism, the formula \( V = \text{length} \times \text{width} \times \text{height} \) is applied. Given the dimensions—depth (length) of 34.25 in., width of 32.62 in., and height of 69.37 in.—the calculation yields a volume of approximately 77,502.59 cubic inches. Option A (2,262.85) is far too small, indicating a miscalculation. Option C (136.24) is implausibly low, likely resulting from using incorrect units or dimensions. Option D (25,834.20) is also incorrect, as it does not reflect the correct multiplication of the given dimensions. Thus, only option B accurately represents the calculated volume.
2^3 * 27^(1/3) * 1^3
- A. 54
- B. 24
- C. 72
- D. 18
Correct Answer & Rationale
Correct Answer: B
To solve the expression \(2^3 \times 27^{(1/3)} \times 1^3\), we first simplify each component. Calculating \(2^3\) gives \(8\). Next, \(27^{(1/3)}\) equals \(3\) since the cube root of \(27\) is \(3\). Finally, \(1^3\) remains \(1\). Now, multiplying these values together: \(8 \times 3 \times 1 = 24\). Option A (54) results from incorrect multiplication. Option C (72) miscalculates the values, and Option D (18) stems from misunderstanding the cube root. Thus, \(24\) is the correct outcome.
To solve the expression \(2^3 \times 27^{(1/3)} \times 1^3\), we first simplify each component. Calculating \(2^3\) gives \(8\). Next, \(27^{(1/3)}\) equals \(3\) since the cube root of \(27\) is \(3\). Finally, \(1^3\) remains \(1\). Now, multiplying these values together: \(8 \times 3 \times 1 = 24\). Option A (54) results from incorrect multiplication. Option C (72) miscalculates the values, and Option D (18) stems from misunderstanding the cube root. Thus, \(24\) is the correct outcome.