Simplify The Expression: ${ \frac{3 A^4 \times (10 A)^3}{(5 A 2) 3} }$

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Introduction

Simplifying algebraic expressions is a crucial skill in mathematics, and it requires a deep understanding of the underlying rules and concepts. In this article, we will focus on simplifying a specific expression involving exponents and variables. The given expression is $\frac{3 a^4 \times (10 a)^3}{(5 a^2)^3}$, and our goal is to simplify it to its simplest form.

Understanding Exponents and Variables

Before we dive into simplifying the expression, let's review the basics of exponents and variables. An exponent is a small number that is written to the upper right of a variable or a number, indicating how many times the variable or number should be multiplied by itself. For example, in the expression $a^3$, the exponent 3 indicates that the variable $a$ should be multiplied by itself three times: $a \times a \times a$.

A variable is a letter or symbol that represents a value that can change. In the expression $a^3$, the variable $a$ represents a value that can be any real number.

Simplifying the Expression

Now that we have a good understanding of exponents and variables, let's simplify the given expression. To simplify the expression, we need to apply the rules of exponents and variables.

The expression is $\frac{3 a^4 \times (10 a)^3}{(5 a^2)^3}$. We can start by simplifying the terms inside the parentheses using the rule of exponents that states $(ab)^n = a^n \times b^n$.

Applying this rule, we get:

$\frac{3 a^4 \times (10 a)^3}{(5 a^2)^3} = \frac{3 a^4 \times 10^3 \times a^3}{5^3 \times a^6}$

Applying the Rule of Exponents

Now that we have simplified the terms inside the parentheses, we can apply the rule of exponents that states $\frac{a^m}{a^n} = a^{m-n}$.

Applying this rule, we get:

$\frac{3 a^4 \times 10^3 \times a^3}{5^3 \times a^6} = \frac{3 \times 10^3 \times a^{4+3}}{5^3 \times a^6}$

Simplifying the Exponents

Now that we have applied the rule of exponents, we can simplify the exponents by combining like terms.

The expression becomes:

$\frac{3 \times 10^3 \times a^{7}}{5^3 \times a^6}$

Canceling Out Common Factors

Now that we have simplified the exponents, we can cancel out common factors between the numerator and the denominator.

The expression becomes:

$\frac{3 \times 10^3 \times a}{5^3}$

Simplifying the Numerator and Denominator

Now that we have canceled out common factors, we can simplify the numerator and denominator by applying the rule of exponents that states $a^m \times a^n = a^{m+n}$.

Applying this rule, we get:

$\frac{3 \times 10^3 \times a}{5^3} = \frac{3 \times 10^3 \times a}{5^3} = \frac{3000a}{125}$

Final Simplification

Now that we have simplified the numerator and denominator, we can simplify the expression further by dividing the numerator and denominator by their greatest common factor.

The expression becomes:

$\frac{3000a}{125} = 24a$

Conclusion

In this article, we simplified the expression $\frac{3 a^4 \times (10 a)^3}{(5 a^2)^3}$ to its simplest form. We applied the rules of exponents and variables to simplify the expression, and we canceled out common factors between the numerator and the denominator. The final simplified expression is $24a$.

Frequently Asked Questions

• What is the rule of exponents that states $(ab)^n = a^n \times b^n$? The rule of exponents that states $(ab)^n = a^n \times b^n$ is a fundamental rule in algebra that allows us to simplify expressions involving exponents and variables.
• What is the rule of exponents that states $\frac{a^m}{a^n} = a^{m-n}$? The rule of exponents that states $\frac{a^m}{a^n} = a^{m-n}$ is a fundamental rule in algebra that allows us to simplify expressions involving exponents and variables.
• What is the rule of exponents that states $a^m \times a^n = a^{m+n}$? The rule of exponents that states $a^m \times a^n = a^{m+n}$ is a fundamental rule in algebra that allows us to simplify expressions involving exponents and variables.

References

• [1] Algebra, 2nd edition, by Michael Artin
• [2] Calculus, 2nd edition, by Michael Spivak
• [3] Mathematics, 2nd edition, by Richard Courant

Further Reading

• Simplifying Algebraic Expressions
• Rules of Exponents
• Variables and Constants

Related Articles

• Simplifying the Expression: $\frac{2x^3 \times (3x)^2}{(4x)^2}$
• Simplifying the Expression: $\frac{5y^4 \times (2y)^3}{(3y)^3}$
• Simplifying the Expression: $\frac{7z^5 \times (4z)^2}{(5z)^2}$
  

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Introduction

In our previous article, we simplified the expression $\frac{3 a^4 \times (10 a)^3}{(5 a^2)^3}$ to its simplest form. We applied the rules of exponents and variables to simplify the expression, and we canceled out common factors between the numerator and the denominator. The final simplified expression is $24a$.

In this article, we will answer some frequently asked questions related to the simplification of the expression.

Q&A

Q: What is the rule of exponents that states $(ab)^n = a^n \times b^n$?

A: The rule of exponents that states $(ab)^n = a^n \times b^n$ is a fundamental rule in algebra that allows us to simplify expressions involving exponents and variables. This rule states that when we have a product of two variables raised to a power, we can separate the variables and raise each one to the power.

Q: What is the rule of exponents that states $\frac{a^m}{a^n} = a^{m-n}$?

A: The rule of exponents that states $\frac{a^m}{a^n} = a^{m-n}$ is a fundamental rule in algebra that allows us to simplify expressions involving exponents and variables. This rule states that when we have a fraction with the same base raised to different powers, we can subtract the exponents.

Q: What is the rule of exponents that states $a^m \times a^n = a^{m+n}$?

A: The rule of exponents that states $a^m \times a^n = a^{m+n}$ is a fundamental rule in algebra that allows us to simplify expressions involving exponents and variables. This rule states that when we have a product of two variables raised to the same power, we can add the exponents.

Q: How do I simplify an expression with exponents and variables?

A: To simplify an expression with exponents and variables, you need to apply the rules of exponents and variables. First, simplify the terms inside the parentheses using the rule of exponents that states $(ab)^n = a^n \times b^n$. Then, apply the rule of exponents that states $\frac{a^m}{a^n} = a^{m-n}$ to simplify the fraction. Finally, apply the rule of exponents that states $a^m \times a^n = a^{m+n}$ to simplify the product.

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. A constant is a value that does not change. For example, in the expression $a^3$, the variable $a$ represents a value that can be any real number, while the constant 3 represents a fixed value.

Q: How do I cancel out common factors between the numerator and the denominator?

A: To cancel out common factors between the numerator and the denominator, you need to identify the common factors and divide them out. For example, in the expression $\frac{3 \times 10^3 \times a}{5^3}$, the common factors are 3 and $5^3$. You can cancel out these factors by dividing them out.

Q: What is the final simplified expression?

A: The final simplified expression is $24a$.

Conclusion

In this article, we answered some frequently asked questions related to the simplification of the expression $\frac{3 a^4 \times (10 a)^3}{(5 a^2)^3}$. We applied the rules of exponents and variables to simplify the expression, and we canceled out common factors between the numerator and the denominator. The final simplified expression is $24a$.

Frequently Asked Questions

• What is the rule of exponents that states $(ab)^n = a^n \times b^n$?
• What is the rule of exponents that states $\frac{a^m}{a^n} = a^{m-n}$?
• What is the rule of exponents that states $a^m \times a^n = a^{m+n}$?
• How do I simplify an expression with exponents and variables?
• What is the difference between a variable and a constant?
• How do I cancel out common factors between the numerator and the denominator?

References

• [1] Algebra, 2nd edition, by Michael Artin
• [2] Calculus, 2nd edition, by Michael Spivak
• [3] Mathematics, 2nd edition, by Richard Courant

Further Reading

• Simplifying Algebraic Expressions
• Rules of Exponents
• Variables and Constants

Related Articles

• Simplifying the Expression: $\frac{2x^3 \times (3x)^2}{(4x)^2}$
• Simplifying the Expression: $\frac{5y^4 \times (2y)^3}{(3y)^3}$
• Simplifying the Expression: $\frac{7z^5 \times (4z)^2}{(5z)^2}$