Question

find the least common multiple of the expressions.
12(x² - 4), 2x(x + 2)
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1. - / 1 points

combine and simplify.
9/(4x) - 8/9
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Explanation:

First Problem: Find the least common multiple of \( 12(x^2 - 4) \) and \( 2x(x + 2) \)

Step 1: Factor the expressions

First, factor \( x^2 - 4 \) using the difference of squares formula \( a^2 - b^2=(a + b)(a - b) \). So \( x^2 - 4=(x + 2)(x - 2) \). Then the first expression \( 12(x^2 - 4) \) becomes \( 12(x + 2)(x - 2) \). We can also factor \( 12 \) into prime factors: \( 12 = 2^2\times3 \). The second expression is \( 2x(x + 2) \), and \( 2 \) is already a prime factor, \( x \) is a linear factor, and \( (x + 2) \) is a linear factor.

Step 2: Identify the highest powers of all factors

For the prime factors:

• The factor \( 2 \): in \( 12(x + 2)(x - 2) \) we have \( 2^2 \), in \( 2x(x + 2) \) we have \( 2^1 \). So we take \( 2^2 \).
• The factor \( 3 \): only present in \( 12(x + 2)(x - 2) \) with power \( 3^1 \), so we take \( 3^1 \).

For the variable and polynomial factors:

• The factor \( x \): only present in \( 2x(x + 2) \) with power \( x^1 \), so we take \( x^1 \).
• The factor \( (x + 2) \): present in both expressions, in \( 12(x + 2)(x - 2) \) with power \( (x + 2)^1 \) and in \( 2x(x + 2) \) with power \( (x + 2)^1 \), so we take \( (x + 2)^1 \).
• The factor \( (x - 2) \): only present in \( 12(x + 2)(x - 2) \) with power \( (x - 2)^1 \), so we take \( (x - 2)^1 \).

Step 3: Multiply the highest powers together

Now, multiply all these highest - power factors: \( 2^2\times3\times x\times(x + 2)\times(x - 2) \)
Calculate \( 2^2\times3=4\times3 = 12 \). Then the product is \( 12x(x + 2)(x - 2) \). We can also expand \( (x + 2)(x - 2)=x^2 - 4 \), so \( 12x(x^2 - 4)=12x^3-48x \), but it's better to leave it in factored form for the least common multiple.

Step 1: Find a common denominator

To subtract two fractions, we need a common denominator. The denominators are \( 4x \) and \( 9 \). Since \( 4x \) and \( 9 \) have no common factors (except 1), the least common denominator (LCD) is the product of the two denominators, which is \( 4x\times9 = 36x \).

Step 2: Rewrite the fractions with the LCD

Rewrite \( \frac{9}{4x} \) with denominator \( 36x \): multiply the numerator and denominator by \( 9 \), so \( \frac{9\times9}{4x\times9}=\frac{81}{36x} \).
Rewrite \( \frac{8}{9} \) with denominator \( 36x \): multiply the numerator and denominator by \( 4x \), so \( \frac{8\times4x}{9\times4x}=\frac{32x}{36x} \).

Step 3: Subtract the fractions

Now that the fractions have the same denominator, we subtract the numerators: \( \frac{81}{36x}-\frac{32x}{36x}=\frac{81 - 32x}{36x} \).

Answer:

\( 12x(x + 2)(x - 2) \) (or \( 12x^3-48x \))

Second Problem: Combine and simplify \( \frac{9}{4x}-\frac{8}{9} \)