Equivalent Expressions MCQ Quiz - Objective Question with Answer for Equivalent Expressions - Download Free PDF

To find \(b\), we start by factoring \(80p^6 - 40p^4\). The GCF is \(40p^4\), so we can write the expression as \(40p^4(2p^2 - 1)\).

We want this to be equivalent to \(b p^4 (4p^2 - 2)\).

Setting these expressions equal gives:

- \(40p^4(2p^2 - 1) = b p^4 (4p^2 - 2)\)

By comparing the coefficients, we get:

- \(40 = b\)

Thus, the value of \(b\) is \(80\).

To find the area of a rectangle, multiply the length and width: \((2a^3b^{-2})(3a^{-1}b^4)\).

1. Multiply the coefficients: \(2 \times 3 = 6\).

2. For \(a\): \(3 + (-1) = 2\), so \(a^2\).

3. For \(b\): \(-2 + 4 = 2\), so \(b^2\).

The area is \(6a^2b^2\). Thus, Option 4 is correct. Other options fail to properly apply exponent rules or miscalculate the coefficients.

The area of the garden is calculated by multiplying the height and width: \((4x^2y^{-3})(5x^{-4}y^5)\).

1. Multiply the coefficients: \(4 \times 5 = 20\).

2. For \(x\): \(2 + (-4) = -2\), hence \(x^{-2}\).

3. For \(y\): \(-3 + 5 = 2\), hence \(y^2\).

Therefore, the area is \(20x^{-2}y^2\), which can be expressed as \(\frac{20y^2}{x^2}\) in terms of positive exponents. Option 1 is correct. Other options either incorrectly add exponents or incorrectly multiply coefficients.

To simplify \( (a^{1/3} \cdot b^{1/4})^{12} \) , apply the rule \((x^m \cdot y^n)^p = x^{mp} \cdot y^{np}\). This gives \((a^{1/3} \cdot b^{1/4})^{12} = a^{(1/3) \cdot 12} \cdot b^{(1/4) \cdot 12}\) . Simplifying the exponents, we get \( a^4 \cdot b^{3} \) . Thus, \( x = 4 \)  and \( y = 3 \) . Therefore, \( x + y = 4 + 3 = 7\) . The correct answer is 7 .

To solve \((a^3b^{-2}c)^2 \cdot (a^{-1}b^4c^3)\), we first apply the power of a power rule to \((a^3b^{-2}c)^2\). This gives us:

\(a^{3 \times 2}b^{-2 \times 2}c^{1 \times 2} = a^6b^{-4}c^2\).

Next, multiply this result by \((a^{-1}b^4c^3)\):

\(a^6b^{-4}c^2 \cdot a^{-1}b^4c^3\).

Using the product of powers rule, we add the exponents for each base:

- For \(a\): \(a^{6 + (-1)} = a^5\).

- For \(b\): \(b^{-4 + 4} = b^0\), and since any number to the power of zero is 1, \(b^0\) simplifies to 1.

- For \(c\): \(c^{2 + 3} = c^5\).

Thus, the simplified expression is \(a^5b^0c^5 = a^5c^5\). Therefore, the correct answer is option 3.

Choice C is correct. The first expression (1.5x - 2.4)² can be rewritten as (1.5x - 2.4)(1.5x - 2.4). Applying the distributive property to this product yields (2.25x² - 3.6x - 3.6x + 5.76)-(5.2x² - 6.4). This difference can be rewritten as (2.25x² - 3.6x - 3.6x + 5.76) + (-1)(5.2x² - 6.4). Distributing the factor of -1 through the second expression yields 2.25x² - 3.6x - 3.6x + 5.76 - 5.2x² + 6.4. Regrouping like terms, the expression becomes (2.25x² - 5.2x²) + (-3.6x - 3.6x) + (5.76 + 6.4). Combining like terms yields -2.95x² - 7.2x + 12.16.

Choices A, B, and D are incorrect and likely result from errors made when applying the distributive property or combining the resulting like terms.

The correct answer is \(\frac{14}{3}\). It's given that 3 teaspoons is equivalent to 1 tablespoon. Therefore, 14 teaspoons is equivalent to \(\left(14 \text { teaspoons) }\left(\frac{1 \text { tablespoon }}{3 \text { teaspoons }}\right) \text {, or } \frac{14}{3}\right.\)teaspoons. Note that 14/3, 4.666, and 4.667 are examples of ways to enter a correct answer.

The correct answer is 73,920. It's given that 1 furlong = 220 yards and 1 yard = 3 feet. It follows that a distance of 112 furlongs is equivalent to \((112 \text { furlongs })\left(\frac{220 \text { yards }}{1 \text { furlong }}\right)\left(\frac{3 \text { foet }}{1 \text { yard }}\right)\), or 73,920 feet.

The correct answer is 40,260. It's given that 1 furlong = 220 yards and 1 yard = 3 feet. It follows that a distance of 61 furlongs is equivalent to \((61 \text { furlongs })\left(\frac{220 \text { yards }}{1 \text { furlong }}\right)\left(\frac{3 \text { foet }}{1 \text { yard }}\right)\). or 40,260 feet.

Choice A is correct. Since 1 kilometer is equal to 1,000 meters, it follows that 90 kilometers is equal to 90(1,000)=90,000  meters. Since 1 hour is equal to 60 minutes and 1 minute is equal to 60 seconds, it follows that 1 hour is equal to 60(60) = 3,600 seconds. Now is equal to \(\frac{90 \text { kilometers }}{1 \text { hour }} \text { is equal to } \frac{90,000 \text { meters }}{3,600 \text { seconds }}\), which reduces to \(\frac{25 \text { meters }}{1 \text { second }}\) or 25 meters per second. 
Choices B, C, and D are incorrect and may result from conceptual or calculation errors. 

Choice C is correct. It's given that 3 teaspoons is equivalent to 1 tablespoon. Therefore, 44 tablespoons is equivalent to \(\left(44 \text { tablespoons) }\left(\frac{3 \text { teaspoons }}{1 \text { tableapoon }}\right)\right.\), or 132 teaspoons.
Choice A is incorrect. This is equivalent to approximately 15.66 tablespoons, not 44 tablespoons.
Choice B is incorrect. This is equivalent to approximately 29.33 tablespoons, not 44 tablespoons.
Choice D is incorrect. This is equivalent to approximately 58.66 tablespoons, not 44 tablespoons.

Choice A is correct. It's given that t = 4u. Multiplying both sides of this equation by 2 yields 2t = 2(4u), or 2t = 8u.
Choice B is incorrect and may result from dividing, instead of multiplying, the right-hand side of the equation by 2. Choices C and D are incorrect and may result from calculation errors.

The correct answer is 102. It's given that 1 yard is equivalent to 3 feet. Therefore, 34 yards is equivalent to \((34 \text { yards })\left(\frac{3 \text { feet }}{1 \text { yard }}\right)\). or 102 feet.

Choice B is correct. It's given that 1 yard = 36 inches. Therefore, 612 inches is equivalent to \(612 \text { inches }\left(\frac{1 \text { yard }}{38 \text { inches }}\right),\) which can be rewritten as \(\frac{612 \text { yards }}{36} \text {, }\)or 17 yards. 36
Choice A is incorrect. This is the number of yards that are equivalent to 2.124 inches.
Choice C is incorrect. This is the number of yards that are equivalent to 20,736 inches.
Choice D is incorrect. This is the number of yards that are equivalent to 793,152 inches.

Choice A is correct. It's given that \(D=T-\frac{9}{25}(100-H)\). Solving this formula for H expresses the relative humidity in terms of the temperature and the dew point. Subtracting T from both sides of this equation yields \(D-T=-\frac{9}{25}(100-H)\). Multiplying both sides by \(-\frac{25}{9}\) yields \(-\frac{25}{9}(D-T)=100-H\). Subtracting 100 from both sides yields \(-\frac{25}{9}(D-T)-100=-H\). Multiplying both sides by -1 results in the formula \(\frac{25}{9}(D-T)+100=H\).

Choices B, C, and D are incorrect and may result from errors made when rewriting the given formula.