Simplify The Expressions.$[ \begin{tabular}{|c|} \hline 3 D − 10 ÷ 6 D − 7 3 D^{-10} \div 6 D^{-7} 3 D − 10 ÷ 6 D − 7 \ \hline 9 D − 5 ⋅ ( 36 D 6 ÷ 4 D 2 ) 9 D^{-5} \cdot\left(36 D^6 \div 4 D^2\right) 9 D − 5 ⋅ ( 36 D 6 ÷ 4 D 2 ) \ \hline 2 D 12 ⋅ D − 2 D 6 2 D^{12} \cdot \frac{d^{-2}}{d^6} 2 D 12 ⋅ D 6 D − 2 ​ \ \hline 125 D 4 ÷ 5 D − 8 125 D^4 \div 5 D^{-8} 125 D 4 ÷ 5 D − 8

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Introduction

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Algebraic manipulations are an essential part of mathematics, and simplifying expressions is a crucial skill that every student and professional should possess. In this article, we will delve into the world of algebra and explore three different expressions that require simplification. We will use the rules of exponents and algebraic properties to simplify each expression and provide a clear understanding of the process.

Expression 1: $3 d^{-10} \div 6 d^{-7}$

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The first expression we will simplify is $3 d^{-10} \div 6 d^{-7}$. To simplify this expression, we need to apply the rules of exponents and division.

Step 1: Apply the Quotient Rule

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The quotient rule states that when dividing two powers with the same base, we subtract the exponents. In this case, we have:

$$\frac{3 d^{-10}}{6 d^{-7}} = \frac{3}{6} \cdot d^{-10-(-7)}$$

Step 2: Simplify the Coefficient

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The coefficient $\frac{3}{6}$ can be simplified by dividing both the numerator and the denominator by their greatest common divisor, which is 3.

$$\frac{3}{6} = \frac{1}{2}$$

Step 3: Simplify the Exponent

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Now, we can simplify the exponent by subtracting the exponents:

$$d^{-10-(-7)} = d^{-10+7} = d^{-3}$$

Step 4: Write the Final Answer

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Combining the simplified coefficient and exponent, we get:

$$\frac{3 d^{-10}}{6 d^{-7}} = \frac{1}{2} d^{-3}$$

Expression 2: $9 d^{-5} \cdot\left(36 d^6 \div 4 d^2\right)$

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The second expression we will simplify is $9 d^{-5} \cdot\left(36 d^6 \div 4 d^2\right)$. To simplify this expression, we need to apply the rules of exponents and multiplication.

Step 1: Apply the Quotient Rule

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First, we need to simplify the expression inside the parentheses by applying the quotient rule:

$$36 d^6 \div 4 d^2 = \frac{36}{4} \cdot d^{6-2} = 9 d^4$$

Step 2: Apply the Product Rule

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Now, we can apply the product rule, which states that when multiplying two powers with the same base, we add the exponents:

$$9 d^{-5} \cdot 9 d^4 = 9 \cdot 9 \cdot d^{-5+4} = 81 d^{-1}$$

Step 3: Write the Final Answer

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The final answer is:

$$9 d^{-5} \cdot\left(36 d^6 \div 4 d^2\right) = 81 d^{-1}$$

Expression 3: $2 d^{12} \cdot \frac{d^{-2}}{d^6}$

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The third expression we will simplify is $2 d^{12} \cdot \frac{d^{-2}}{d^6}$. To simplify this expression, we need to apply the rules of exponents and division.

Step 1: Apply the Quotient Rule

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First, we need to simplify the fraction by applying the quotient rule:

$$\frac{d^{-2}}{d^6} = d^{-2-6} = d^{-8}$$

Step 2: Apply the Product Rule

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Now, we can apply the product rule:

$$2 d^{12} \cdot d^{-8} = 2 \cdot d^{12-8} = 2 d^4$$

Step 3: Write the Final Answer

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The final answer is:

$$2 d^{12} \cdot \frac{d^{-2}}{d^6} = 2 d^4$$

Expression 4: $125 d^4 \div 5 d^{-8}$

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The fourth expression we will simplify is $125 d^4 \div 5 d^{-8}$. To simplify this expression, we need to apply the rules of exponents and division.

Step 1: Apply the Quotient Rule

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First, we need to simplify the expression by applying the quotient rule:

$$125 d^4 \div 5 d^{-8} = \frac{125}{5} \cdot d^{4-(-8)} = 25 d^{4+8} = 25 d^{12}$$

Step 2: Write the Final Answer

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The final answer is:

$$125 d^4 \div 5 d^{-8} = 25 d^{12}$$

Conclusion

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In this article, we have simplified four different expressions using the rules of exponents and algebraic properties. We have applied the quotient rule, product rule, and simplified coefficients and exponents to arrive at the final answers. These expressions are essential in mathematics and are used in various fields, including physics, engineering, and computer science. By mastering the skills of simplifying expressions, you will be able to tackle complex problems and arrive at accurate solutions.

Frequently Asked Questions

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Q: What is the quotient rule in algebra?

A: The quotient rule states that when dividing two powers with the same base, we subtract the exponents.

Q: What is the product rule in algebra?

A: The product rule states that when multiplying two powers with the same base, we add the exponents.

Q: How do I simplify an expression with a negative exponent?

A: To simplify an expression with a negative exponent, we can rewrite it as a positive exponent by changing the sign of the exponent.

Q: What is the difference between a coefficient and an exponent?

A: A coefficient is a number that is multiplied by a variable, while an exponent is a power to which a variable is raised.

Final Thoughts

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Simplifying expressions is an essential skill in mathematics, and mastering it will help you tackle complex problems and arrive at accurate solutions. By applying the rules of exponents and algebraic properties, you can simplify expressions and arrive at the final answers. Remember to always follow the order of operations and simplify coefficients and exponents to arrive at the final answer.

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Q&A: Simplifying Expressions

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In this article, we will continue to explore the world of algebra and provide answers to frequently asked questions about simplifying expressions.

Q: What is the quotient rule in algebra?

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A: The quotient rule states that when dividing two powers with the same base, we subtract the exponents. For example, $\frac{a^m}{a^n} = a^{m-n}$.

Q: What is the product rule in algebra?

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A: The product rule states that when multiplying two powers with the same base, we add the exponents. For example, $a^m \cdot a^n = a^{m+n}$.

Q: How do I simplify an expression with a negative exponent?

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A: To simplify an expression with a negative exponent, we can rewrite it as a positive exponent by changing the sign of the exponent. For example, $a^{-m} = \frac{1}{a^m}$.

Q: What is the difference between a coefficient and an exponent?

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A: A coefficient is a number that is multiplied by a variable, while an exponent is a power to which a variable is raised. For example, in the expression $3x^2$, the coefficient is 3 and the exponent is 2.

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

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A: To simplify an expression with multiple variables, we can use the rules of exponents and algebraic properties. For example, if we have the expression $2x^2y^3 \cdot 3x^4y^2$, we can simplify it by combining like terms and applying the product rule.

Q: What is the order of operations in algebra?

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A: The order of operations in algebra is:

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

Q: How do I simplify an expression with a fraction?

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A: To simplify an expression with a fraction, we can use the rules of exponents and algebraic properties. For example, if we have the expression $\frac{2x^2}{3x^4}$, we can simplify it by canceling out common factors and applying the quotient rule.

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

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A: A rational expression is an expression that can be written as a fraction, while an irrational expression is an expression that cannot be written as a fraction. For example, $\frac{2x^2}{3x^4}$ is a rational expression, while $\sqrt{2x^2}$ is an irrational expression.

Q: How do I simplify an expression with a square root?

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A: To simplify an expression with a square root, we can use the rules of exponents and algebraic properties. For example, if we have the expression $\sqrt{2x^2}$, we can simplify it by rewriting it as $x\sqrt{2}$.

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

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A: A linear expression is an expression that can be written in the form $ax + b$, while a quadratic expression is an expression that can be written in the form $ax^2 + bx + c$. For example, $2x + 3$ is a linear expression, while $x^2 + 2x + 1$ is a quadratic expression.

Q: How do I simplify an expression with a linear factor?

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A: To simplify an expression with a linear factor, we can use the rules of exponents and algebraic properties. For example, if we have the expression $(x + 2)(x - 3)$, we can simplify it by multiplying the factors together.

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

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A: A polynomial expression is an expression that can be written in the form $a_nx^n + a_{n-1}x^{n-1} + \cdots + a_1x + a_0$, while a non-polynomial expression is an expression that cannot be written in this form. For example, $x^2 + 2x + 1$ is a polynomial expression, while $\sqrt{x^2 + 2x + 1}$ is a non-polynomial expression.

Q: How do I simplify an expression with a non-polynomial factor?

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A: To simplify an expression with a non-polynomial factor, we can use the rules of exponents and algebraic properties. For example, if we have the expression $(x^2 + 2x + 1)\sqrt{x^2 + 2x + 1}$, we can simplify it by rewriting it as $(x^2 + 2x + 1)^{\frac{3}{2}}$.

Q: What is the difference between a real number and an imaginary number?

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A: A real number is a number that can be written in the form $a + bi$, where $a$ and $b$ are real numbers and $i$ is the imaginary unit, while an imaginary number is a number that can be written in the form $bi$, where $b$ is a real number and $i$ is the imaginary unit. For example, $3 + 4i$ is a complex number, while $4i$ is an imaginary number.

Q: How do I simplify an expression with a complex number?

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A: To simplify an expression with a complex number, we can use the rules of exponents and algebraic properties. For example, if we have the expression $(3 + 4i)(2 - 5i)$, we can simplify it by multiplying the complex numbers together.

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

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A: A rational function is a function that can be written as a ratio of two polynomials, while an irrational function is a function that cannot be written as a ratio of two polynomials. For example, $\frac{x^2 + 2x + 1}{x^2 - 2x + 1}$ is a rational function, while $\sqrt{x^2 + 2x + 1}$ is an irrational function.

Q: How do I simplify an expression with a rational function?

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A: To simplify an expression with a rational function, we can use the rules of exponents and algebraic properties. For example, if we have the expression $\frac{x^2 + 2x + 1}{x^2 - 2x + 1}$, we can simplify it by canceling out common factors and applying the quotient rule.

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

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A: A linear equation is an equation that can be written in the form $ax + b = c$, while a quadratic equation is an equation that can be written in the form $ax^2 + bx + c = 0$. For example, $2x + 3 = 5$ is a linear equation, while $x^2 + 2x + 1 = 0$ is a quadratic equation.

Q: How do I solve a linear equation?

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A: To solve a linear equation, we can use the rules of algebra and the properties of equality. For example, if we have the equation $2x + 3 = 5$, we can solve it by subtracting 3 from both sides and then dividing both sides by 2.

Q: How do I solve a quadratic equation?

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A: To solve a quadratic equation, we can use the quadratic formula, which is $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$. For example, if we have the equation $x^2 + 2x + 1 = 0$, we can solve it by using the quadratic formula.

Q: What is the difference between a system of linear equations and a system of quadratic equations?

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A: A system of linear equations is a set of equations that can be written in the form $ax + by = c$, while a system of quadratic equations is a set of equations that can be written in the form $ax^2 + bx + c = 0$. For example, $\begin{cases} 2x + 3y = 5 \\ x - 2y = -3 \end{cases}$ is a system of linear equations, while $\begin{cases} x^2 + 2x + 1 = 0 \\ x^2 - 2x + 1 = 0 \end{cases}$ is a system of quadratic equations.

Q: How do I solve a system of linear equations?

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A: To solve a system of linear equations, we can use the method of substitution or the method of elimination. For example, if we have the system $\begin{cases} 2x + 3y = 5 \\ x - 2y = -3 \end{cases}$, we can solve it by using the method of substitution.

Q: How do I solve a system of quadratic equations?

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A: To solve a system of quadratic equations, we can use the method of substitution or the method of elimination. For example, if we have the system $\begin{cases} x^2 + 2x + 1 = 0