Simplify The Following Expressions:1. $\left(x 5\right) 7$2. $\frac{21 X^7 Y^4}{7 X^2 Y^2}$3. $3^9 \cdot 3^{-6}$4. \left(6 A^{10} C^9\right)\left(a B^{-5}\right)\left(5 B^{12} C^6\right ]

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Algebraic expressions are a fundamental concept in mathematics, and simplifying them is an essential skill for any math enthusiast. In this article, we will explore four different algebraic expressions and simplify them using various techniques. We will cover the basics of exponent rules, fraction simplification, and product of powers.

Simplifying Exponential Expressions

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1. Simplifying $\left(x^5\right)^7$

To simplify the expression $\left(x^5\right)^7$, we need to apply the power rule of exponents, which states that $\left(a^m\right)^n = a^{m \cdot n}$. Using this rule, we can rewrite the expression as:

$$\left(x^5\right)^7 = x^{5 \cdot 7} = x^{35}$$

Therefore, the simplified expression is $x^{35}$.

2. Simplifying $\frac{21 x^7 y^4}{7 x^2 y^2}$

To simplify the expression $\frac{21 x^7 y^4}{7 x^2 y^2}$, we need to apply the quotient rule of exponents, which states that $\frac{a^m}{a^n} = a^{m - n}$. We can also simplify the fraction by dividing the numerator and denominator by their greatest common factor, which is 7.

$$\frac{21 x^7 y^4}{7 x^2 y^2} = \frac{3 x^7 y^4}{x^2 y^2} = 3 x^{7 - 2} y^{4 - 2} = 3 x^5 y^2$$

Therefore, the simplified expression is $3 x^5 y^2$.

3. Simplifying $3^9 \cdot 3^{-6}$

To simplify the expression $3^9 \cdot 3^{-6}$, we need to apply the product rule of exponents, which states that $a^m \cdot a^n = a^{m + n}$. We can also simplify the expression by combining the exponents.

$$3^9 \cdot 3^{-6} = 3^{9 + (-6)} = 3^3$$

Therefore, the simplified expression is $3^3$.

4. Simplifying $\left(6 a^{10} c^9\right)\left(a b^{-5}\right)\left(5 b^{12} c^6\right)$

To simplify the expression $\left(6 a^{10} c^9\right)\left(a b^{-5}\right)\left(5 b^{12} c^6\right)$, we need to apply the product rule of exponents, which states that $a^m \cdot a^n = a^{m + n}$. We can also simplify the expression by combining the exponents.

$$\left(6 a^{10} c^9\right)\left(a b^{-5}\right)\left(5 b^{12} c^6\right) = 6 \cdot 5 \cdot a^{10 + 1 + 0} \cdot b^{-5 + 12} \cdot c^{9 + 6} = 30 a^{11} b^7 c^{15}$$

Therefore, the simplified expression is $30 a^{11} b^7 c^{15}$.

Conclusion

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Simplifying algebraic expressions is an essential skill for any math enthusiast. By applying the power rule, quotient rule, and product rule of exponents, we can simplify complex expressions and make them easier to work with. In this article, we have explored four different algebraic expressions and simplified them using various techniques. We hope that this article has provided you with a better understanding of how to simplify algebraic expressions and has given you the confidence to tackle more complex math problems.

Frequently Asked Questions

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Q: What is the power rule of exponents?

A: The power rule of exponents states that $\left(a^m\right)^n = a^{m \cdot n}$.

Q: What is the quotient rule of exponents?

A: The quotient rule of exponents states that $\frac{a^m}{a^n} = a^{m - n}$.

Q: What is the product rule of exponents?

A: The product rule of exponents states that $a^m \cdot a^n = a^{m + n}$.

Q: How do I simplify an algebraic expression?

A: To simplify an algebraic expression, you need to apply the power rule, quotient rule, and product rule of exponents. You can also simplify the expression by combining the exponents and dividing the numerator and denominator by their greatest common factor.

Further Reading

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If you want to learn more about simplifying algebraic expressions, we recommend checking out the following resources:

• Khan Academy: Algebraic Expressions
• Mathway: Simplifying Algebraic Expressions
• Wolfram Alpha: Algebraic Expressions

We hope that this article has provided you with a better understanding of how to simplify algebraic expressions and has given you the confidence to tackle more complex math problems. Happy math-ing!

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Algebraic expressions are a fundamental concept in mathematics, and simplifying them is an essential skill for any math enthusiast. In this article, we will answer some of the most frequently asked questions about algebraic expressions and provide you with a better understanding of how to simplify them.

Q&A: Algebraic Expressions

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Q: What is an algebraic expression?

A: An algebraic expression is a mathematical expression that consists of variables, constants, and mathematical operations such as addition, subtraction, multiplication, and division.

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

A: An algebraic expression is a mathematical expression that consists of variables, constants, and mathematical operations, while an equation is a statement that two expressions are equal.

Q: How do I simplify an algebraic expression?

A: To simplify an algebraic expression, you need to apply the power rule, quotient rule, and product rule of exponents. You can also simplify the expression by combining the exponents and dividing the numerator and denominator by their greatest common factor.

Q: What is the power rule of exponents?

A: The power rule of exponents states that $\left(a^m\right)^n = a^{m \cdot n}$.

Q: What is the quotient rule of exponents?

A: The quotient rule of exponents states that $\frac{a^m}{a^n} = a^{m - n}$.

Q: What is the product rule of exponents?

A: The product rule of exponents states that $a^m \cdot a^n = a^{m + n}$.

Q: How do I simplify an expression with negative exponents?

A: To simplify an expression with negative exponents, you need to apply the rule that $a^{-m} = \frac{1}{a^m}$.

Q: How do I simplify an expression with fractional exponents?

A: To simplify an expression with fractional exponents, you need to apply the rule that $a^{\frac{m}{n}} = \sqrt[n]{a^m}$.

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

A: A variable is a symbol that represents a value that can change, while a constant is a value that does not change.

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

A: To simplify an expression with multiple variables, you need to apply the rules of exponents and combine the variables.

Examples of Algebraic Expressions

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Example 1: Simplifying $\left(x^5\right)^7$

To simplify the expression $\left(x^5\right)^7$, we need to apply the power rule of exponents, which states that $\left(a^m\right)^n = a^{m \cdot n}$. Using this rule, we can rewrite the expression as:

$$\left(x^5\right)^7 = x^{5 \cdot 7} = x^{35}$$

Therefore, the simplified expression is $x^{35}$.

Example 2: Simplifying $\frac{21 x^7 y^4}{7 x^2 y^2}$

To simplify the expression $\frac{21 x^7 y^4}{7 x^2 y^2}$, we need to apply the quotient rule of exponents, which states that $\frac{a^m}{a^n} = a^{m - n}$. We can also simplify the fraction by dividing the numerator and denominator by their greatest common factor, which is 7.

$$\frac{21 x^7 y^4}{7 x^2 y^2} = \frac{3 x^7 y^4}{x^2 y^2} = 3 x^{7 - 2} y^{4 - 2} = 3 x^5 y^2$$

Therefore, the simplified expression is $3 x^5 y^2$.

Conclusion

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Algebraic expressions are a fundamental concept in mathematics, and simplifying them is an essential skill for any math enthusiast. By applying the power rule, quotient rule, and product rule of exponents, we can simplify complex expressions and make them easier to work with. In this article, we have answered some of the most frequently asked questions about algebraic expressions and provided you with a better understanding of how to simplify them.

Further Reading

---

If you want to learn more about algebraic expressions, we recommend checking out the following resources:

• Khan Academy: Algebraic Expressions
• Mathway: Simplifying Algebraic Expressions
• Wolfram Alpha: Algebraic Expressions

We hope that this article has provided you with a better understanding of how to simplify algebraic expressions and has given you the confidence to tackle more complex math problems. Happy math-ing!