Use The Following Expression To Answer Parts $A$ And $D$.This Expression Can Be Evaluated Quickly By Doing The Computation In Only One Of The Parentheses:$[ -3\left(\frac{6}{11}+\frac{7}{19}\right)

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Understanding the Problem

When dealing with mathematical expressions that involve parentheses, it's essential to evaluate them correctly to obtain the correct result. In this article, we will focus on a specific expression that can be evaluated quickly by doing the computation in only one of the parentheses. This approach will help us simplify the expression and arrive at the correct solution.

The Given Expression

The expression we need to evaluate is:

−3(611+719){ -3\left(\frac{6}{11}+\frac{7}{19}\right) }

Evaluating the Expression Inside the Parentheses

To evaluate the expression quickly, we need to find a common denominator for the fractions inside the parentheses. The least common multiple (LCM) of 11 and 19 is 209. We can rewrite the fractions with the common denominator as follows:

611=6×1911×19=114209{ \frac{6}{11} = \frac{6 \times 19}{11 \times 19} = \frac{114}{209} }

719=7×1119×11=77209{ \frac{7}{19} = \frac{7 \times 11}{19 \times 11} = \frac{77}{209} }

Adding the Fractions

Now that we have the fractions with a common denominator, we can add them together:

114209+77209=191209{ \frac{114}{209} + \frac{77}{209} = \frac{191}{209} }

Multiplying by -3

Finally, we can multiply the result by -3:

−3×191209=−573209{ -3 \times \frac{191}{209} = \frac{-573}{209} }

Simplifying the Result

We can simplify the result by dividing both the numerator and the denominator by their greatest common divisor (GCD). The GCD of 573 and 209 is 1, so the result is already simplified.

Conclusion

In this article, we demonstrated how to evaluate an expression with parentheses quickly by doing the computation in only one of the parentheses. We found a common denominator for the fractions inside the parentheses, added them together, and then multiplied the result by -3. The final result was −573209\frac{-573}{209}. This approach can be applied to similar expressions to simplify the computation and arrive at the correct solution.

Real-World Applications

Evaluating expressions with parentheses is a fundamental skill in mathematics that has numerous real-world applications. In finance, for example, it's essential to evaluate expressions that involve parentheses to calculate interest rates, investment returns, and other financial metrics. In science, evaluating expressions with parentheses is crucial in calculating physical quantities such as velocity, acceleration, and force.

Tips and Tricks

When dealing with expressions that involve parentheses, here are some tips and tricks to keep in mind:

  • Find a common denominator: When adding or subtracting fractions, it's essential to find a common denominator to ensure that the fractions are equivalent.
  • Simplify the expression: Before evaluating the expression, simplify it by combining like terms and eliminating any unnecessary parentheses.
  • Use the order of operations: When evaluating expressions, follow the order of operations (PEMDAS) to ensure that the expression is evaluated correctly.

Common Mistakes

When evaluating expressions with parentheses, here are some common mistakes to avoid:

  • Forgetting to find a common denominator: When adding or subtracting fractions, forgetting to find a common denominator can lead to incorrect results.
  • Not simplifying the expression: Failing to simplify the expression before evaluating it can lead to unnecessary complexity and errors.
  • Not following the order of operations: Failing to follow the order of operations (PEMDAS) can lead to incorrect results and errors.

Conclusion

Evaluating expressions with parentheses is a fundamental skill in mathematics that has numerous real-world applications. By following the tips and tricks outlined in this article, you can simplify the computation and arrive at the correct solution. Remember to find a common denominator, simplify the expression, and follow the order of operations to ensure that the expression is evaluated correctly.

Q: What is the order of operations when evaluating expressions with parentheses?

A: The order of operations when evaluating expressions with parentheses is:

  1. Parentheses: Evaluate the expressions inside the parentheses first.
  2. Exponents: Evaluate any exponents (such as squaring or cubing) 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 find a common denominator when adding or subtracting fractions?

A: To find a common denominator when adding or subtracting fractions, you need to find the least common multiple (LCM) of the denominators. The LCM is the smallest number that both denominators can divide into evenly.

For example, if you have the fractions 1/2 and 1/3, the LCM of 2 and 3 is 6. So, you can rewrite the fractions as 3/6 and 2/6, and then add them together.

Q: What is the difference between a numerator and a denominator?

A: The numerator is the number on top of a fraction, and the denominator is the number on the bottom. The numerator tells you how many equal parts you have, and the denominator tells you how many parts the whole is divided into.

For example, in the fraction 3/4, the numerator is 3 and the denominator is 4. This means that you have 3 equal parts out of a total of 4 parts.

Q: How do I simplify a fraction?

A: To simplify a fraction, you need to find the greatest common divisor (GCD) of the numerator and the denominator. The GCD is the largest number that both numbers can divide into evenly.

For example, if you have the fraction 12/16, the GCD of 12 and 16 is 4. So, you can divide both the numerator and the denominator by 4 to get the simplified fraction 3/4.

Q: What is the difference between a positive and negative exponent?

A: A positive exponent means that you multiply the base number by itself as many times as the exponent says. For example, 2^3 means 2 x 2 x 2 = 8.

A negative exponent means that you divide 1 by the base number raised to the power of the exponent. For example, 2^-3 means 1 / (2 x 2 x 2) = 1/8.

Q: How do I evaluate an expression with parentheses and exponents?

A: To evaluate an expression with parentheses and exponents, you need to follow the order of operations:

  1. Evaluate the expressions inside the parentheses first.
  2. Evaluate any exponents (such as squaring or cubing) next.
  3. Evaluate any multiplication and division operations from left to right.
  4. Finally, evaluate any addition and subtraction operations from left to right.

For example, if you have the expression (2^3 + 4) / 2, you would evaluate it as follows:

  1. Evaluate the expression inside the parentheses: 2^3 = 8, so (2^3 + 4) = (8 + 4) = 12.
  2. Divide 12 by 2: 12 / 2 = 6.

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 solve a quadratic equation?

A: To solve a quadratic equation, you can use the quadratic formula:

x = (-b ± √(b^2 - 4ac)) / 2a

where a, b, and c are the coefficients of the quadratic equation.

For example, if you have the quadratic equation x^2 + 4x + 4 = 0, you can plug in the values a = 1, b = 4, and c = 4 into the quadratic formula to solve for x.

Q: What is the difference between a rational expression and an irrational expression?

A: A rational expression is an expression that can be written as a fraction of two integers. For example, 3/4 is a rational expression.

An irrational expression is an expression that cannot be written as a fraction of two integers. For example, √2 is an irrational expression.

Q: How do I simplify a rational expression?

A: To simplify a rational expression, you need to find the greatest common divisor (GCD) of the numerator and the denominator. The GCD is the largest number that both numbers can divide into evenly.

For example, if you have the rational expression 12/16, the GCD of 12 and 16 is 4. So, you can divide both the numerator and the denominator by 4 to get the simplified rational expression 3/4.

Q: What is the difference between a polynomial and a non-polynomial expression?

A: A polynomial is an expression that consists of variables and coefficients combined using only addition, subtraction, and multiplication. For example, 2x^2 + 3x - 4 is a polynomial.

A non-polynomial expression is an expression that contains variables and coefficients combined using operations other than addition, subtraction, and multiplication. For example, √x is a non-polynomial expression.

Q: How do I evaluate a polynomial expression?

A: To evaluate a polynomial expression, you need to follow the order of operations:

  1. Evaluate any exponents (such as squaring or cubing) next.
  2. Evaluate any multiplication and division operations from left to right.
  3. Finally, evaluate any addition and subtraction operations from left to right.

For example, if you have the polynomial expression 2x^2 + 3x - 4, you would evaluate it as follows:

  1. Evaluate the exponent: 2x^2 = 2(x x x) = 2x^2.
  2. Multiply 2x^2 by 3x: 2x^2 x 3x = 6x^3.
  3. Subtract 4 from 6x^3: 6x^3 - 4.

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

A: A function is a relation between a set of inputs (called the domain) and a set of possible outputs (called the range). 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 evaluate a function expression?

A: To evaluate a function expression, you need to substitute the input value into the function and simplify the expression.

For example, if you have the function f(x) = 2x + 3 and you want to evaluate f(4), you would substitute x = 4 into the function and simplify the expression:

f(4) = 2(4) + 3 = 8 + 3 = 11

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

A: A linear function is a function that can be written in the form f(x) = mx + b, where m and b are constants. For example, f(x) = 2x + 3 is a linear function.

A non-linear function is a function that cannot be written in the form f(x) = mx + b. For example, f(x) = x^2 + 3x - 4 is a non-linear function.

Q: How do I graph a linear function?

A: To graph a linear function, you need to find the x-intercept and the y-intercept of the function.

The x-intercept is the point where the function crosses the x-axis. To find the x-intercept, set y = 0 and solve for x.

The y-intercept is the point where the function crosses the y-axis. To find the y-intercept, set x = 0 and solve for y.

For example, if you have the linear function f(x) = 2x + 3, you can find the x-intercept by setting y = 0 and solving for x:

0 = 2x + 3 -3 = 2x -3/2 = x

So, the x-intercept is (-3/2, 0).

To find the y-intercept, set x = 0 and solve for y:

y = 2(0) + 3 = 3

So, the y-intercept is (0, 3).

Q: What is the difference between a quadratic function and a non-quadratic function?

A: A quadratic function is a function that can be written in the form f(x) = ax^2 + bx + c, where a, b, and c are constants. For example, f(x) = x^2 +