Simplifying Expressions Practice Questions

To simplify, group the like terms together. The variable terms are \(3x\) and \(2x\), and the constant terms are \(5\) and \(-8\):

\((3x + 2x) + (5 – 8)\)

Combine each group:

\(= 5x – 3\)

First, apply the distributive property to eliminate the parentheses. Multiply each term inside by 4:

\(4(2x – 3) + 5x = 8x – 12 + 5x\)

Then combine like terms:

\(= 13x – 12\)

A common mistake is to forget to distribute the 4 to the second term inside the parentheses, which would incorrectly give \(13x – 3\) (choice A).

Only like terms (terms with the same variable) can be combined. Group the \(a\) terms and the \(b\) terms separately:

\((7a – 2a) + (3b + 5b)\)

\(= 5a + 8b\)

Note that choice B is incorrect because \(a\) and \(b\) are different variables and cannot be combined into a single \(ab\) term.

The negative sign in front of the parentheses means you multiply each term inside by \(-1\):

\(-(3x – 7) + 2x = -3x + 7 + 2x\)

Now combine like terms:

\(= -x + 7\)

A common error is to negate only the first term and leave the 7 negative, which would give \(-x – 7\) (choice A). Remember to distribute the negative sign to every term inside the parentheses.

When multiplying powers with the same base, add the exponents:

\(x^a \cdot x^b = x^{a+b}\)

Therefore:

\(x^3 \cdot x^5 = x^{3+5} = x^8\)

Choice A incorrectly multiplies the exponents (which applies to a power raised to a power, not multiplication). Choice D incorrectly subtracts the exponents (which applies to division).

When raising a product to a power, apply the exponent to every factor inside the parentheses:

\((2x^3)^2 = 2^2 \cdot (x^3)^2\)

Compute each part. For the coefficient: \(2^2 = 4\). For the variable, multiply the exponents:

\((x^3)^2 = x^{3 \times 2} = x^6\)

Putting it together:

\((2x^3)^2 = 4x^6\)

Choice A forgets to square the coefficient 2.

Apply the distributive property to each set of parentheses. Be especially careful with the second group, where the \(-2\) must be distributed to both terms:

\(3(x + 4) – 2(x – 5)\)\(\:= 3x + 12 – 2x + 10\)

Note that \(-2 \times -5 = +10\), not \(-10\). Now combine like terms:

\(= x + 22\)

Choice A results from incorrectly computing \(-2 \times -5\) as \(-10\) instead of \(+10\).

Group the like terms by their degree. Remember that \(x^2\) terms and \(x\) terms are not like terms:

\((5x^2 – 2x^2) + (3x + 7x) + (-4)\)

Combine each group:

\(= 3x^2 + 10x – 4\)

When dividing monomials, divide the coefficients and subtract the exponents for each variable:

\(\dfrac{12}{4} = 3\)

\(x^{5-2} = x^3\)

\(y^{3-1} = y^2\)

Therefore:

\(\dfrac{12x^5y^3}{4x^2y} = 3x^3y^2\)

Choice D incorrectly adds the exponents instead of subtracting them, which would apply to multiplication, not division.

Work from the inside out. First, distribute the 3 to the inner parentheses:

\(2[3(x – 1) + 4] = 2[3x – 3 + 4]\)

Combine the constants inside the brackets:

\(= 2[3x + 1]\)

Finally, distribute the 2:

\(= 6x + 2\)

To simplify a square root, find the largest perfect square factor of the number under the radical. The largest perfect square factor of 50 is 25:

\(\sqrt{50} = \sqrt{25 \times 2}\)

Use the property \(\sqrt{a \times b} = \sqrt{a} \times \sqrt{b}\) to separate the factors:

\(= \sqrt{25} \times \sqrt{2} = 5\sqrt{2}\)

Choice D writes \(2\sqrt{25}\), which is not simplified and equals 10, not \(\sqrt{50}\).

To simplify a rational expression, first factor the numerator. Recognize that \(x^2 – 9\) is a difference of squares:

\(\dfrac{x^2 – 9}{x + 3} = \dfrac{(x + 3)(x – 3)}{x + 3}\)

The common factor of \((x + 3)\) appears in both the numerator and denominator, so it cancels:

\(= x – 3\)

Note that this simplification is valid only when \(x \neq -3\), since the original expression is undefined at that value.

When multiplying powers with the same base, add the exponents — even when one of them is negative:

\(x^{-3} \cdot x^7 = x^{-3 + 7} = x^4\)

Choice B incorrectly multiplies the exponents (\(-3 \times 7 = -21\)), which only applies when raising a power to a power. Choice C adds the absolute values (\(3 + 7 = 10\)) without accounting for the negative sign.

The perimeter of a rectangle is \(P = 2l + 2w\). Substitute the given expressions for length and width:

\(P = 2(3x + 2) + 2(x – 1)\)

Distribute the 2 to each group:

\(= 6x + 4 + 2x – 2\)

Combine like terms:

\(= 8x + 2\)

Start by simplifying the numerator. Apply the exponent of 3 to every factor inside the parentheses:

\((3x^2y)^3\)\(\:= 3^3 \cdot (x^2)^3 \cdot y^3\)\(\:= 27x^6y^3\)

Now divide by the denominator:

\(\dfrac{27x^6y^3}{9x^4y^2}\)

Divide the coefficients and subtract the exponents for each variable:

\(\dfrac{27}{9} = 3\), \(\:x^{6-4} = x^2\), \(\:y^{3-2} = y\)

Therefore:

\(\dfrac{(3x^2y)^3}{9x^4y^2} = 3x^2y\)