Simplify 12 Y 7 18 Y − 3 \frac{12 Y^7}{18 Y^{-3}} 18 Y − 3 12 Y 7 ​ .

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Introduction

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Simplifying algebraic expressions is a crucial skill in mathematics, and it's essential to understand the rules and techniques involved. In this article, we will focus on simplifying a specific expression, $\frac{12 y^7}{18 y^{-3}}$. We will use various techniques, including factoring, canceling, and applying the rules of exponents.

Understanding the Expression

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The given expression is $\frac{12 y^7}{18 y^{-3}}$. To simplify this expression, we need to understand the rules of exponents and how to apply them. The expression consists of two parts: the numerator and the denominator. The numerator is $12 y^7$, and the denominator is $18 y^{-3}$.

Applying the Rules of Exponents

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To simplify the expression, we need to apply the rules of exponents. The first rule we will use is the rule for dividing like bases with exponents. This rule states that when we divide two numbers with the same base and exponents, we subtract the exponents.

Rule 1: Dividing Like Bases with Exponents

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The rule for dividing like bases with exponents is as follows:

$$\frac{a^m}{a^n} = a^{m-n}$$

where $a$ is the base, and $m$ and $n$ are the exponents.

Applying the Rule to the Expression

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We can apply this rule to the expression $\frac{12 y^7}{18 y^{-3}}$. We will divide the numerator and the denominator by their greatest common factor (GCF), which is $6$. This will give us:

$$\frac{12 y^7}{18 y^{-3}} = \frac{2 y^7}{3 y^{-3}}$$

Now, we can apply the rule for dividing like bases with exponents. We will divide the numerator and the denominator by their like bases, which are $y$.

$$\frac{2 y^7}{3 y^{-3}} = \frac{2 y^{7-(-3)}}{3}$$

Simplifying the Exponents

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Now, we can simplify the exponents. When we subtract a negative exponent, we add the absolute value of the exponent. In this case, we have:

$$\frac{2 y^{7-(-3)}}{3} = \frac{2 y^{7+3}}{3}$$

$$\frac{2 y^{7+3}}{3} = \frac{2 y^{10}}{3}$$

Final Simplification

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The final step is to simplify the fraction. We can do this by dividing the numerator and the denominator by their greatest common factor (GCF), which is $1$. This will give us:

$$\frac{2 y^{10}}{3}$$

Conclusion

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In this article, we simplified the expression $\frac{12 y^7}{18 y^{-3}}$ using various techniques, including factoring, canceling, and applying the rules of exponents. We applied the rule for dividing like bases with exponents and simplified the exponents. The final simplified expression is $\frac{2 y^{10}}{3}$.

Frequently Asked Questions

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Q: What is the rule for dividing like bases with exponents?

A: The rule for dividing like bases with exponents is $\frac{a^m}{a^n} = a^{m-n}$, where $a$ is the base, and $m$ and $n$ are the exponents.

Q: How do I simplify exponents when subtracting a negative exponent?

A: When subtracting a negative exponent, you add the absolute value of the exponent. For example, $7-(-3) = 7+3 = 10$.

Q: What is the final simplified expression for $\frac{12 y^7}{18 y^{-3}}$?

A: The final simplified expression is $\frac{2 y^{10}}{3}$.

Further Reading

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

• Algebraic Expressions
• Rules of Exponents
• Simplifying Algebraic Expressions

References

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• Algebra
• Exponentiation
• Simplifying Algebraic Expressions
  

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Introduction

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In our previous article, we simplified the expression $\frac{12 y^7}{18 y^{-3}}$ using various techniques, including factoring, canceling, and applying the rules of exponents. In this article, we will answer some frequently asked questions (FAQs) related to simplifying algebraic expressions.

Q&A

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Q: What is the rule for dividing like bases with exponents?

A: The rule for dividing like bases with exponents is $\frac{a^m}{a^n} = a^{m-n}$, where $a$ is the base, and $m$ and $n$ are the exponents.

Q: How do I simplify exponents when subtracting a negative exponent?

A: When subtracting a negative exponent, you add the absolute value of the exponent. For example, $7-(-3) = 7+3 = 10$.

Q: What is the final simplified expression for $\frac{12 y^7}{18 y^{-3}}$?

A: The final simplified expression is $\frac{2 y^{10}}{3}$.

Q: Can I simplify an expression with a variable in the exponent?

A: Yes, you can simplify an expression with a variable in the exponent. For example, $\frac{2 x^7}{3 x^{-3}} = \frac{2 x^{7-(-3)}}{3} = \frac{2 x^{7+3}}{3} = \frac{2 x^{10}}{3}$.

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

A: To simplify an expression with a negative exponent, you can rewrite the expression with a positive exponent. For example, $\frac{2 x^{-3}}{3} = \frac{2}{3 x^3}$.

Q: Can I simplify an expression with a fraction in the exponent?

A: Yes, you can simplify an expression with a fraction in the exponent. For example, $\frac{2 x^{3/4}}{3} = \frac{2}{3} x^{3/4}$.

Q: How do I simplify an expression with a variable in the denominator?

A: To simplify an expression with a variable in the denominator, you can multiply the numerator and the denominator by the conjugate of the denominator. For example, $\frac{2 x}{x+1} = \frac{2 x(x-1)}{(x+1)(x-1)} = \frac{2 x^2 - 2 x}{x^2 - 1}$.

Tips and Tricks

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Tip 1: Simplify the expression inside the parentheses first

When simplifying an expression, it's often helpful to simplify the expression inside the parentheses first. For example, $\frac{2 (x+1)}{x+1} = 2$.

Tip 2: Use the rule for dividing like bases with exponents

When dividing like bases with exponents, use the rule $\frac{a^m}{a^n} = a^{m-n}$.

Tip 3: Simplify the exponents before simplifying the fraction

When simplifying an expression, it's often helpful to simplify the exponents before simplifying the fraction. For example, $\frac{2 x^{10}}{3} = \frac{2 x^{10}}{3}$.

Conclusion

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In this article, we answered some frequently asked questions (FAQs) related to simplifying algebraic expressions. We also provided some tips and tricks for simplifying expressions. Remember to simplify the expression inside the parentheses first, use the rule for dividing like bases with exponents, and simplify the exponents before simplifying the fraction.

Further Reading

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

• Algebraic Expressions
• Rules of Exponents
• Simplifying Algebraic Expressions

References

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• Algebra
• Exponentiation
• Simplifying Algebraic Expressions