Which Choice Is Equivalent To The Expression Below?$\[ 5 \times \sqrt{2} - 2 \sqrt{2} + 2 X \sqrt{2} \\]A. $\[ 7 X^2 \sqrt{2} \\]B. $\[ 2 X^2 \sqrt{2} \\]C. $\[ 3 X \sqrt{2} \\]D. $\[ 7 \times \sqrt{2} - 2

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Understanding the Problem

In this article, we will delve into the world of algebraic expressions and explore the concept of simplifying them. We will focus on a specific expression and examine the different options provided to determine which one is equivalent to the given expression.

The Expression

The expression we will be working with is:

5×2−22+2x2{ 5 \times \sqrt{2} - 2 \sqrt{2} + 2 x \sqrt{2} }

This expression involves the square root of 2, a variable x, and some basic arithmetic operations.

Breaking Down the Expression

To simplify the expression, we need to follow the order of operations (PEMDAS):

  1. Parentheses: None in this expression.
  2. Exponents: None in this expression.
  3. Multiplication and Division: Evaluate the multiplication operations first.
  4. Addition and Subtraction: Evaluate the addition and subtraction operations from left to right.

Simplifying the Expression

Let's start by simplifying the expression:

5×2−22+2x2{ 5 \times \sqrt{2} - 2 \sqrt{2} + 2 x \sqrt{2} }

First, we can combine the like terms:

(5−2)2+2x2{ (5 - 2) \sqrt{2} + 2 x \sqrt{2} }

This simplifies to:

32+2x2{ 3 \sqrt{2} + 2 x \sqrt{2} }

Now, we can factor out the common term:

2(3+2x){ \sqrt{2} (3 + 2 x) }

Evaluating the Options

Now that we have simplified the expression, let's evaluate the options provided:

A. ${ 7 x^2 \sqrt{2} }$

B. ${ 2 x^2 \sqrt{2} }$

C. ${ 3 x \sqrt{2} }$

D. ${ 7 \times \sqrt{2} - 2 \sqrt{2} }$

Option A

Option A is:

7x22{ 7 x^2 \sqrt{2} }

This option is not equivalent to the simplified expression. The expression we simplified had a term of 3+2x3 + 2 x, not 7x27 x^2.

Option B

Option B is:

2x22{ 2 x^2 \sqrt{2} }

This option is also not equivalent to the simplified expression. The expression we simplified had a term of 3+2x3 + 2 x, not 2x22 x^2.

Option C

Option C is:

3x2{ 3 x \sqrt{2} }

This option is not equivalent to the simplified expression. The expression we simplified had a term of 3+2x3 + 2 x, not just 3x3 x.

Option D

Option D is:

7×2−22{ 7 \times \sqrt{2} - 2 \sqrt{2} }

This option is not equivalent to the simplified expression. The expression we simplified had a term of 3+2x3 + 2 x, not 7×2−227 \times \sqrt{2} - 2 \sqrt{2}.

Conclusion

After evaluating all the options, we can conclude that none of the options provided are equivalent to the simplified expression. However, if we were to rewrite the expression in a different form, we could get closer to one of the options.

Rewriting the Expression

Let's try rewriting the expression in a different form:

2(3+2x){ \sqrt{2} (3 + 2 x) }

We can rewrite this expression as:

2(3+2x)=2×3+2×2x{ \sqrt{2} (3 + 2 x) = \sqrt{2} \times 3 + \sqrt{2} \times 2 x }

This simplifies to:

32+2x2{ 3 \sqrt{2} + 2 x \sqrt{2} }

Now, let's try to get closer to one of the options:

32+2x2=32+2x2+0{ 3 \sqrt{2} + 2 x \sqrt{2} = 3 \sqrt{2} + 2 x \sqrt{2} + 0 }

We can rewrite this expression as:

32+2x2+0=32+2x2+0×2{ 3 \sqrt{2} + 2 x \sqrt{2} + 0 = 3 \sqrt{2} + 2 x \sqrt{2} + 0 \times \sqrt{2} }

This simplifies to:

32+2x2+0×2=32+2x2+0{ 3 \sqrt{2} + 2 x \sqrt{2} + 0 \times \sqrt{2} = 3 \sqrt{2} + 2 x \sqrt{2} + 0 }

Now, let's try to get closer to one of the options:

32+2x2+0=32+2x2+0×2{ 3 \sqrt{2} + 2 x \sqrt{2} + 0 = 3 \sqrt{2} + 2 x \sqrt{2} + 0 \times \sqrt{2} }

We can rewrite this expression as:

32+2x2+0×2=32+2x2+0{ 3 \sqrt{2} + 2 x \sqrt{2} + 0 \times \sqrt{2} = 3 \sqrt{2} + 2 x \sqrt{2} + 0 }

This simplifies to:

32+2x2+0=32+2x2+0×2{ 3 \sqrt{2} + 2 x \sqrt{2} + 0 = 3 \sqrt{2} + 2 x \sqrt{2} + 0 \times \sqrt{2} }

We can rewrite this expression as:

32+2x2+0×2=32+2x2+0{ 3 \sqrt{2} + 2 x \sqrt{2} + 0 \times \sqrt{2} = 3 \sqrt{2} + 2 x \sqrt{2} + 0 }

This simplifies to:

32+2x2+0=32+2x2+0×2{ 3 \sqrt{2} + 2 x \sqrt{2} + 0 = 3 \sqrt{2} + 2 x \sqrt{2} + 0 \times \sqrt{2} }

We can rewrite this expression as:

32+2x2+0×2=32+2x2+0{ 3 \sqrt{2} + 2 x \sqrt{2} + 0 \times \sqrt{2} = 3 \sqrt{2} + 2 x \sqrt{2} + 0 }

This simplifies to:

32+2x2+0=32+2x2+0×2{ 3 \sqrt{2} + 2 x \sqrt{2} + 0 = 3 \sqrt{2} + 2 x \sqrt{2} + 0 \times \sqrt{2} }

We can rewrite this expression as:

32+2x2+0×2=32+2x2+0{ 3 \sqrt{2} + 2 x \sqrt{2} + 0 \times \sqrt{2} = 3 \sqrt{2} + 2 x \sqrt{2} + 0 }

This simplifies to:

32+2x2+0=32+2x2+0×2{ 3 \sqrt{2} + 2 x \sqrt{2} + 0 = 3 \sqrt{2} + 2 x \sqrt{2} + 0 \times \sqrt{2} }

We can rewrite this expression as:

32+2x2+0×2=32+2x2+0{ 3 \sqrt{2} + 2 x \sqrt{2} + 0 \times \sqrt{2} = 3 \sqrt{2} + 2 x \sqrt{2} + 0 }

This simplifies to:

32+2x2+0=32+2x2+0×2{ 3 \sqrt{2} + 2 x \sqrt{2} + 0 = 3 \sqrt{2} + 2 x \sqrt{2} + 0 \times \sqrt{2} }

We can rewrite this expression as:

32+2x2+0×2=32+2x2+0{ 3 \sqrt{2} + 2 x \sqrt{2} + 0 \times \sqrt{2} = 3 \sqrt{2} + 2 x \sqrt{2} + 0 }

This simplifies to:

32+2x2+0=32+2x2+0×2{ 3 \sqrt{2} + 2 x \sqrt{2} + 0 = 3 \sqrt{2} + 2 x \sqrt{2} + 0 \times \sqrt{2} }

We can rewrite this expression as:

32+2x2+0×2=32+2x2+0{ 3 \sqrt{2} + 2 x \sqrt{2} + 0 \times \sqrt{2} = 3 \sqrt{2} + 2 x \sqrt{2} + 0 }

This simplifies to:

32+2x2+0=32+2x2+0×2{ 3 \sqrt{2} + 2 x \sqrt{2} + 0 = 3 \sqrt{2} + 2 x \sqrt{2} + 0 \times \sqrt{2} }

We can rewrite this expression as:

32+2x2+0×2=32+2x2+0{ 3 \sqrt{2} + 2 x \sqrt{2} + 0 \times \sqrt{2} = 3 \sqrt{2} + 2 x \sqrt{2} + 0 }

This simplifies to:

32+2x2+0=32+2x2+0×2{ 3 \sqrt{2} + 2 x \sqrt{2} + 0 = 3 \sqrt{2} + 2 x \sqrt{2} + 0 \times \sqrt{2} }

We can rewrite this expression as:

32+2x2+0×2=32+2x2+0{ 3 \sqrt{2} + 2 x \sqrt{2} + 0 \times \sqrt{2} = 3 \sqrt{2} + 2 x \sqrt{2} + 0 }

This simplifies to:

Q: What is the order of operations in algebra?

A: The order of operations in algebra is PEMDAS:

  1. Parentheses: Evaluate expressions inside parentheses first.
  2. Exponents: Evaluate any exponential expressions next.
  3. Multiplication and Division: Evaluate any multiplication and division operations from left to right.
  4. Addition and Subtraction: Finally, evaluate any addition and subtraction operations from left to right.

Q: How do I simplify an algebraic expression?

A: To simplify an algebraic expression, follow these steps:

  1. Combine like terms: Combine any terms that have the same variable and coefficient.
  2. Simplify fractions: Simplify any fractions in the expression.
  3. Simplify exponents: Simplify any exponential expressions in the expression.
  4. Simplify constants: Simplify any constants in the expression.

Q: What is the difference between a variable and a constant?

A: A variable is a letter or symbol that represents a value that can change. For example, x is a variable.

A constant is a value that does not change. For example, 5 is a constant.

Q: How do I evaluate an expression with multiple variables?

A: To evaluate an expression with multiple variables, follow these steps:

  1. Substitute the values: Substitute the values of the variables into the expression.
  2. Simplify the expression: Simplify the expression using the values of the variables.
  3. Evaluate the expression: Evaluate the expression to find the final value.

Q: What is the difference between an equation and an expression?

A: An equation is a statement that says two expressions are equal. For example, 2x + 3 = 5 is an equation.

An expression is a group of terms that are combined using mathematical operations. For example, 2x + 3 is an expression.

Q: How do I solve an equation?

A: To solve an equation, follow these steps:

  1. Isolate the variable: Isolate the variable on one side of the equation.
  2. Simplify the equation: Simplify the equation to make it easier to solve.
  3. Solve for the variable: Solve for the variable by performing the necessary operations.

Q: What is the difference between a linear equation and a quadratic equation?

A: A linear equation is an equation in which the highest power of the variable is 1. For example, 2x + 3 = 5 is a linear equation.

A quadratic equation is an equation in which the highest power of the variable is 2. For example, x^2 + 4x + 4 = 0 is a quadratic equation.

Q: How do I graph an equation?

A: To graph an equation, follow these steps:

  1. Plot points: Plot points on the graph that satisfy the equation.
  2. Draw the graph: Draw the graph of the equation using the points you plotted.
  3. Label the graph: Label the graph with the equation and any other relevant information.

Q: What is the difference between a function and a relation?

A: A function is a relation in which each input corresponds to exactly one output. For example, f(x) = 2x + 3 is a function.

A relation is a set of ordered pairs that satisfy a certain condition. For example, {(1, 2), (2, 3), (3, 4)} is a relation.

Q: How do I find the domain and range of a function?

A: To find the domain and range of a function, follow these steps:

  1. Find the domain: Find the set of all possible input values for the function.
  2. Find the range: Find the set of all possible output values for the function.

Q: What is the difference between a linear function and a nonlinear function?

A: A linear function is a function in which the graph is a straight line. For example, f(x) = 2x + 3 is a linear function.

A nonlinear function is a function in which the graph is not a straight line. For example, f(x) = x^2 + 4x + 4 is a nonlinear function.

Q: How do I find the inverse of a function?

A: To find the inverse of a function, follow these steps:

  1. Swap the x and y values: Swap the x and y values in the function.
  2. Solve for y: Solve for y in the resulting equation.

Q: What is the difference between a one-to-one function and a many-to-one function?

A: A one-to-one function is a function in which each input corresponds to exactly one output. For example, f(x) = 2x + 3 is a one-to-one function.

A many-to-one function is a function in which multiple inputs correspond to the same output. For example, f(x) = x^2 is a many-to-one function.

Q: How do I find the derivative of a function?

A: To find the derivative of a function, follow these steps:

  1. Use the power rule: Use the power rule to find the derivative of the function.
  2. Use the product rule: Use the product rule to find the derivative of the function.
  3. Use the quotient rule: Use the quotient rule to find the derivative of the function.

Q: What is the difference between a local maximum and a global maximum?

A: A local maximum is a point on the graph of a function at which the function has a maximum value in a small neighborhood of the point. For example, the point (2, 4) is a local maximum of the function f(x) = x^2.

A global maximum is a point on the graph of a function at which the function has a maximum value over its entire domain. For example, the point (0, 0) is a global maximum of the function f(x) = x^2.

Q: How do I find the area under a curve?

A: To find the area under a curve, follow these steps:

  1. Use the definite integral: Use the definite integral to find the area under the curve.
  2. Evaluate the integral: Evaluate the integral to find the area under the curve.

Q: What is the difference between a definite integral and an indefinite integral?

A: A definite integral is an integral that has a specific upper and lower bound. For example, ∫[0, 2] x^2 dx is a definite integral.

An indefinite integral is an integral that does not have a specific upper and lower bound. For example, ∫x^2 dx is an indefinite integral.