For all non-zero real numbers a, b, and c, the expression below is simplified: \frac{(2a^2… — ACT

For all non-zero real numbers

a

b

, and

c

, the expression below is simplified:

\frac{(2a^2 b^{-1} c^3)^3}{4a^5 (b^2 c^{-2})^{-2}}

Which of the following is equivalent to this expression?

A
$\frac{2ac^5}{b^7}$
B
$\frac{abc^5}{2}$
$2abc^5$ Answer
D
$4abc^5$
E
$2b^2c^{10}$

Answer

The expression is equivalent to

2abc^5

The correct answer is obtained by first simplifying the numerator and denominator using the power of a product and power of a power rules, then dividing the coefficients and subtracting the exponents of like bases. This yields the simplified expression

2abc^5

Step-by-Step Solution

Apply the power of a product rule to the numerator.

(2a^2 b^{-1} c^3)^3 = 2^3 \cdot (a^2)^3 \cdot (b^{-1})^3 \cdot (c^3)^3 = 8 a^6 b^{-3} c^9

Each factor inside the parentheses must be raised to the power of 3, multiplying the exponents of the variables.

Apply the power of a product rule to the denominator's parentheses.

4a^5 (b^2 c^{-2})^{-2} = 4a^5 \cdot (b^2)^{-2} \cdot (c^{-2})^{-2} = 4a^5 b^{-4} c^4

The terms inside the parentheses are raised to the power of -2, multiplying their exponents.

Divide the simplified numerator by the simplified denominator.

2a^1 b^1 c^5

, which is

2abc^5

Divide the coefficients (8 / 4 = 2) and subtract the exponents of the same bases:

a^{6-5} = a^1

b^{-3 - (-4)} = b^1

, and

c^{9-4} = c^5

Key Concept

Properties of Exponents in Algebraic Expressions

Alternative Method

Alternatively, you can rewrite the expression by eliminating negative exponents first. Convert

b^{-1}

\frac{1}{b}

b^2

to itself, and

c^{-2}

\frac{1}{c^2}

within the parentheses, apply the outer exponents, and then simplify the resulting complex fraction.

Estimated Time:1m 30s

Question

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Alternative Method

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