Find the LCM of 6, 2xy2, and 12x2y.

Question

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:
Find the LCM of 6, $2xy^2$ , and $12x^2y$ .

Find the prime factorization of each term.
$6 = 2 \times 3$ $2xy^2 = 2 \times x \times y^2$ $12x^2y = 2^2 \times 3 \times x^2 \times y$
Identify the highest power of each prime factor and variable present in any of the terms.
- For the prime factor 2: The highest power is $2^2$ (from $12x^2y$ ).
- For the prime factor 3: The highest power is $3^1$ (from 6 and $12x^2y$ ).
- For the variable $x$ : The highest power is $x^2$ (from $12x^2y$ ).
- For the variable $y$ : The highest power is $y^2$ (from $2xy^2$ ).
Multiply these highest powers together to find the LCM.
$\text{LCM} = 2^2 \times 3 \times x^2 \times y^2 = 4 \times 3 \times x^2 \times y^2 = 12x^2y^2$
The correct option is C.
$\boxed{\text{C. } 12x^2y^2}$
:
Find $x$ in the equation $\frac{2x}{3} + \frac{x}{4} = 6$ .
Find a common denominator for the fractions.
The least common multiple of 3 and 4 is 12.
Multiply both sides of the equation by 12 to eliminate the denominators.
$12 \left( \frac{2x}{3} + \frac{x}{4} \right) = 12(6)$ $12 \left( \frac{2x}{3} \right) + 12 \left( \frac{x}{4} \right) = 72$ $4(2x) + 3(x) = 72$
Simplify and solve for $x$ .
$8x + 3x = 72$ $11x = 72$ $x = \frac{72}{11}$
To express this as a mixed number: $72 \div 11 = 6$ with a remainder of 6.
$x = 6\frac{6}{11}$
The correct option is C.
$\boxed{\text{C. } 6\frac{6}{11}}$
:
Factorize the expression $xy - x + y - 1$ .
Group the terms.
$(xy - x) + (y - 1)$
Factor out the common factor from each group.
From the first group $(xy - x)$ , factor out $x$ : $x(y - 1)$ .
The second group is already $(y - 1)$ .
$x(y - 1) + 1(y - 1)$
Factor out the common binomial factor $(y - 1)$ .
$(y - 1)(x + 1)$
The correct option is A.
$\boxed{\text{A. } (x+1)(y-1)}$
:
Factorize $a^2 + 8a + 16$ .
Recognize the expression as a perfect square trinomial.
A perfect square trinomial has the form $x^2 + 2xy + y^2 = (x+y)^2$ .
Here, $a^2$ is the first term, and 16 is $4^2$ .
The middle term $8a$ is $2 \times a \times 4$ .
Write the expression in factored form.
$a^2 + 8a + 16 = (a + 4)^2$
The correct option is C.
$\boxed{\text{C. } (a+4)^2}$
:
Simplify $74^2 - 26^2$ .
Use the difference of squares formula, $a^2 - b^2 = (a - b)(a + b)$ .
Here, $a = 74$ and $b = 26$ .
$74^2 - 26^2 = (74 - 26)(74 + 26)$
Perform the subtraction and addition.
$74 - 26 = 48$ $74 + 26 = 100$
Multiply the results.
$48 \times 100 = 4800$
The correct option is A.
$\boxed{\text{A. } 4800}$
:
Using the graph above, for what range of $x$ is $y < 0$ ?
Identify the points where the graph intersects the x-axis (where $y=0$ ).
From the graph, the parabola crosses the x-axis at $x = -2$ and $x = 3$ .
Determine the region where the graph is below the x-axis.
The parabola is below the x-axis between its x-intercepts. This means for values of $x$ greater than $-2$ and less than 3.
Express this range as an inequality.
$-2 < x < 3$
The correct option is A.
$\boxed{\text{A. } -2 < x < 3}$

Find the LCM of 6, 2xy2, and 12x2y.

Answer

Step-by-step solution

Find the prime factorization of each term.

Identify the highest power of each prime factor and variable present in any of the terms.

Multiply these highest powers together to find the LCM.

Find a common denominator for the fractions.

Simplify and solve for $x$ .

Group the terms.

Factor out the common factor from each group.

Factor out the common binomial factor $(y - 1)$ .

Recognize the expression as a perfect square trinomial.

Write the expression in factored form.

Use the difference of squares formula, $a^2 - b^2 = (a - b)(a + b)$ .

Perform the subtraction and addition.

Multiply the results.

Identify the points where the graph intersects the x-axis (where $y=0$ ).

Determine the region where the graph is below the x-axis.

Express this range as an inequality.

Try one yourself

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More Mathematics Questions

Find the LCM of 6, 2xy2, and 12x2y.

Answer

Step-by-step solution

Find the prime factorization of each term.

Identify the highest power of each prime factor and variable present in any of the terms.

Multiply these highest powers together to find the LCM.

Find a common denominator for the fractions.

Simplify and solve for xxx.

Group the terms.

Factor out the common factor from each group.

Factor out the common binomial factor (y−1)(y - 1)(y−1).

Recognize the expression as a perfect square trinomial.

Write the expression in factored form.

Use the difference of squares formula, a2−b2=(a−b)(a+b)a^2 - b^2 = (a - b)(a + b)a2−b2=(a−b)(a+b).

Perform the subtraction and addition.

Multiply the results.

Identify the points where the graph intersects the x-axis (where y=0y=0y=0).

Determine the region where the graph is below the x-axis.

Express this range as an inequality.

Try one yourself

Still stuck on this one?

More Mathematics Questions

Simplify and solve for $x$ .

Factor out the common binomial factor $(y - 1)$ .

Use the difference of squares formula, $a^2 - b^2 = (a - b)(a + b)$ .

Identify the points where the graph intersects the x-axis (where $y=0$ ).