Manipulate Algebraic Expressions - Praxis Core Math

Which of the following is equivalent to $p - (3 - p) + (p - 6)$ ?
Answer: $3p - 9$ To simplify the expression, first distribute the negative sign to the terms in the first parenthesis: $p - 3 + p$ . The second parenthesis can be written without the parenthesis: $+ p - 6$ . The full expression is now $p - 3 + p + p - 6$ . Combine the like terms (the 'p' terms and the constant terms): $(p+p+p) + (-3-6)$ . This simplifies to $3p - 9$ .
The expression for the surface area of a cylinder is $2\pi r^2 + 2\pi rh$ . Which of the following represents this expression in factored form?
Answer: $2\pi r(r+h)$ To factor the expression $2\pi r^2 + 2\pi rh$ , we need to find the greatest common factor (GCF) of the two terms. Both terms contain the factors 2, $\pi$ , and $r$ . Therefore, the GCF is $2\pi r$ . Factoring this out from each term, we get $2\pi r(r) + 2\pi r(h)$ . This simplifies to $2\pi r(r+h)$ .
A large rectangular field has a length of $5x+4$ and a width of $2x-1$ . A smaller rectangular patch within the field has dimensions $x+2$ and $x-1$ . Which expression represents the area of the field that is outside the smaller patch?
Answer: $9x^2 + 2x - 2$ First, find the area of the large field: $A_{large} = (5x+4)(2x-1) = 10x^2 - 5x + 8x - 4 = 10x^2 + 3x - 4$ . Next, find the area of the smaller patch: $A_{small} = (x+2)(x-1) = x^2 - x + 2x - 2 = x^2 + x - 2$ . To find the remaining area, subtract the small area from the large area: $(10x^2 + 3x - 4) - (x^2 + x - 2)$ . Distribute the negative sign: $10x^2 + 3x - 4 - x^2 - x + 2$ . Combine like terms: $(10x^2-x^2) + (3x-x) + (-4+2)$ , which simplifies to $9x^2 + 2x - 2$ .
For $x \neq 2$ and $x \neq -2$ , the expression $\frac{x^2 + x - 6}{x^2 - 4}$ can be simplified to which of the following?
Answer: $\frac{x + 3}{x + 2}$ To simplify the rational expression, first factor the numerator and the denominator. The numerator $x^2 + x - 6$ factors into $(x+3)(x-2)$ . The denominator $x^2 - 4$ is a difference of squares and factors into $(x+2)(x-2)$ . The expression becomes $\frac{(x+3)(x-2)}{(x+2)(x-2)}$ . Cancel the common factor $(x-2)$ from the numerator and denominator. The simplified expression is $\frac{x+3}{x+2}$ .
Which of the following expressions is equivalent to $3(x^2 - 2y + 5) - (x^2 + 4y - 6)$ ?
Answer: $2x^2 - 10y + 21$ First, distribute the 3 into the first parenthesis: $3x^2 - 6y + 15$ . Next, distribute the negative sign into the second parenthesis: $-x^2 - 4y + 6$ . Now, combine the resulting expressions: $(3x^2 - 6y + 15) + (-x^2 - 4y + 6)$ . Combine like terms: $(3x^2 - x^2) + (-6y - 4y) + (15 + 6)$ , which simplifies to $2x^2 - 10y + 21$ .
The area of a square is given by the expression $9x^2 + 30x + 25$ . Which expression represents the length of one side of the square?
Answer: $3x + 5$ The area of a square is the side length squared ( $A = s^2$ ). To find the side length, we need to find the square root of the area expression, which means we need to factor the perfect square trinomial $9x^2 + 30x + 25$ . A perfect square trinomial has the form $a^2 + 2ab + b^2 = (a+b)^2$ . Here, $a^2 = 9x^2$ so $a=3x$ , and $b^2 = 25$ so $b=5$ . Check the middle term: $2ab = 2(3x)(5) = 30x$ . This matches. Thus, the factored form is $(3x+5)^2$ . The length of one side is $3x+5$ .
Which of the following expressions is equivalent to $(64a^6)^{1/2} - (8a^9)^{1/3}$ for $a \ge 0$ ?
Answer: $6a^3$ Simplify each term separately. For the first term: $(64a^6)^{1/2} = \sqrt{64} \cdot \sqrt{a^6} = 8 \cdot a^3 = 8a^3$ . For the second term: $(8a^9)^{1/3} = \sqrt[3]{8} \cdot \sqrt[3]{a^9} = 2 \cdot a^3 = 2a^3$ . The expression becomes $8a^3 - 2a^3$ . Combine like terms: $8a^3 - 2a^3 = 6a^3$ .
If $x = -4$ , what is the value of the expression $x^2 - 3(x+5)$ ?
Answer: $13$ Substitute $x = -4$ into the expression: $(-4)^2 - 3((-4)+5)$ . Following the order of operations, first calculate the value inside the parentheses: $(-4)+5 = 1$ . Next, evaluate the exponent: $(-4)^2 = 16$ . The expression is now $16 - 3(1)$ . Perform the multiplication: $3(1)=3$ . Finally, perform the subtraction: $16 - 3 = 13$ .
Simplify the expression $\sqrt{16x^2} + \sqrt{9x^2}$ for $x \ge 0$ .
Answer: $7x$ First, simplify each square root term individually. $\sqrt{16x^2} = \sqrt{16} \cdot \sqrt{x^2} = 4x$ (since $x \ge 0$ ). Similarly, $\sqrt{9x^2} = \sqrt{9} \cdot \sqrt{x^2} = 3x$ . The expression becomes $4x + 3x$ . Since these are like terms, they can be added together to get $7x$ . It is incorrect to add the numbers under the radicals first.
The perimeter of a triangle is given by the expression $7x-2$ . If the lengths of two sides are $2x+5$ and $3x-4$ , what is the length of the third side?
Answer: $2x - 3$ The perimeter of a triangle is the sum of its three sides. Let the third side be $S$ . Then $P = (2x+5) + (3x-4) + S$ . We are given $P = 7x-2$ . First, find the sum of the two known sides: $(2x+5) + (3x-4) = 5x+1$ . Now, subtract this sum from the total perimeter to find the length of the third side: $S = (7x-2) - (5x+1)$ . Distribute the negative: $S = 7x - 2 - 5x - 1$ . Combine like terms: $S = (7x-5x) + (-2-1) = 2x-3$ .
The expression $(x+y)^{-2}$ is equivalent to which of the following?
Answer: $\frac{1}{(x+y)^2}$ A negative exponent indicates the reciprocal of the base raised to the positive value of the exponent. In this case, the base is $(x+y)$ and the exponent is $-2$ . Therefore, $(x+y)^{-2} = \frac{1}{(x+y)^2}$ . It is a common mistake to distribute the exponent to each term inside the parenthesis, but this is incorrect.
Which expression is equivalent to $ab - 5a + 2b - 10$ ?
Answer: $(a+2)(b-5)$ This expression can be factored by grouping. Group the first two terms and the last two terms: $(ab - 5a) + (2b - 10)$ . Factor out the greatest common factor from each group: $a(b - 5) + 2(b - 5)$ . Now, factor out the common binomial factor $(b - 5)$ : $(a + 2)(b - 5)$ .
Which expression is equivalent to $(x^2+3x-1) - (x^2-x+4) + (2x^2+5)$ ?
Answer: $2x^2 + 4x$ To simplify, first remove the parentheses, being careful to distribute the negative sign. The expression becomes $x^2+3x-1 - x^2+x-4 + 2x^2+5$ . Now, group and combine like terms. For $x^2$ terms: $x^2 - x^2 + 2x^2 = 2x^2$ . For $x$ terms: $3x + x = 4x$ . For constant terms: $-1 - 4 + 5 = 0$ . The simplified expression is $2x^2 + 4x$ .
Which of the following expressions is equivalent to $\frac{18x^5 - 12x^4 + 6x^2}{6x^2}$ for $x \neq 0$ ?
Answer: $3x^3 - 2x^2 + 1$ To simplify, divide each term in the numerator by the monomial in the denominator. $\frac{18x^5}{6x^2} - \frac{12x^4}{6x^2} + \frac{6x^2}{6x^2}$ . For the first term, $18/6 = 3$ and $x^5/x^2 = x^{5-2} = x^3$ . For the second term, $12/6 = 2$ and $x^4/x^2 = x^{4-2} = x^2$ . For the third term, $6x^2/6x^2 = 1$ . Combining these results gives $3x^3 - 2x^2 + 1$ .
If $x^2 + kx - 18$ can be factored into $(x-3)(x+6)$ , what is the value of $k$ ?
Answer: $3$ To find the value of $k$ , we need to expand the factored form $(x-3)(x+6)$ and compare it to the original trinomial $x^2 + kx - 18$ . Using the FOIL method on $(x-3)(x+6)$ : First terms $x \cdot x = x^2$ , Outer terms $x \cdot 6 = 6x$ , Inner terms $-3 \cdot x = -3x$ , and Last terms $-3 \cdot 6 = -18$ . Combining these gives $x^2 + 6x - 3x - 18$ , which simplifies to $x^2 + 3x - 18$ . Comparing this to $x^2 + kx - 18$ , we can see that $k$ must be 3.
The expression $x^4 - 81$ is equivalent to which of the following?
Answer: $(x^2 + 9)(x - 3)(x + 3)$ The expression $x^4 - 81$ is a difference of squares, where the terms are $(x^2)^2$ and $9^2$ . It factors into $(x^2 - 9)(x^2 + 9)$ . The term $x^2 + 9$ is a sum of squares and cannot be factored further using real numbers. The term $x^2 - 9$ is another difference of squares, which factors into $(x - 3)(x + 3)$ . Therefore, the completely factored expression is $(x - 3)(x + 3)(x^2 + 9)$ .
A rectangle's length is 3 feet more than twice its width, $w$ . Which expression represents the area of the rectangle in square feet?
Answer: $2w^2 + 3w$ Let $w$ be the width of the rectangle. The length, $l$ , is described as '3 feet more than twice its width,' which translates to the expression $l = 2w + 3$ . The area of a rectangle is given by the formula $A = l \times w$ . Substituting the expression for the length, we get $A = (2w + 3)w$ . Using the distributive property, this simplifies to $A = 2w^2 + 3w$ .
Which of the following is equivalent to $\frac{2}{x-1} + \frac{3}{x+1}$ ?
Answer: $\frac{5x-1}{x^2-1}$ To add these fractions, find a common denominator, which is $(x-1)(x+1)$ or $x^2-1$ . Rewrite each fraction with the common denominator: $\frac{2(x+1)}{(x-1)(x+1)} + \frac{3(x-1)}{(x-1)(x+1)}$ . Now, add the numerators: $\frac{2(x+1) + 3(x-1)}{x^2-1}$ . Distribute in the numerator: $\frac{2x+2+3x-3}{x^2-1}$ . Combine like terms in the numerator: $\frac{5x-1}{x^2-1}$ .
If $a - b = 5$ and $a + b = 11$ , what is the value of the expression $a^2 - b^2$ ?
Answer: $55$ This problem can be solved without finding the individual values of $a$ and $b$ . The expression $a^2 - b^2$ is a difference of squares, which factors into $(a - b)(a + b)$ . We are given the values of these two factors: $a - b = 5$ and $a + b = 11$ . Therefore, $a^2 - b^2 = (5)(11) = 55$ .
Which of the following expressions is equivalent to $(2p - 5q)^2$ ?
Answer: $4p^2 - 20pq + 25q^2$ To expand $(2p - 5q)^2$ , we multiply the binomial by itself: $(2p - 5q)(2p - 5q)$ . Using the FOIL method: First terms $(2p)(2p)=4p^2$ , Outer terms $(2p)(-5q)=-10pq$ , Inner terms $(-5q)(2p)=-10pq$ , and Last terms $(-5q)(-5q)=25q^2$ . Combining these gives $4p^2 - 10pq - 10pq + 25q^2$ , which simplifies to $4p^2 - 20pq + 25q^2$ .

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