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: $(2a^3{b}^{-2})(3a^{-1}{b}^4)$.

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 $6a^2{b}^2$. 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: $(4x^2{y}^{-3})(5x^{-4}{y}^5)$.

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

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

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

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

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 $a^5{b}^0{c}^5=a^5{c}^5$. 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 $(14\text{ teaspoons) }\left(\frac{1\text{ tablespoon }}{3\text{ teaspoons }}\right)\text{, or }\frac{14}{3}$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 $(44\text{ tablespoons) }\left(\frac{3\text{ teaspoons }}{1\text{ tableapoon }}\right)$, 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.