Finish Simplifying The Expression. Write Your Answer Without Negative Exponents.$\[ \begin{aligned} \left(\frac{x^6 Y^{-9}}{x^{-3} Y^6}\right)^{1 / 3} & =\frac{x^{6 \cdot 1 / 3} Y^{-9 \cdot 1 / 3}}{x^{-3 \cdot 1 / 3} Y^{6 \cdot 1 / 3}} \\ &

Feb 26, 2025 by ADMIN 241 views

**Simplifying Exponential Expressions: A Step-by-Step Guide**

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Understanding Exponents and Negative Exponents

Exponents are a fundamental concept in mathematics, used to represent repeated multiplication of a number. A negative exponent, on the other hand, represents the reciprocal of a number raised to a positive exponent. In this article, we will focus on simplifying an exponential expression that contains negative exponents.

The Expression to Simplify

The given expression is:

$\left(\frac{x^6 y^{-9}}{x^{-3} y^6}\right)^{1 / 3}$

Our goal is to simplify this expression without using negative exponents.

Step 1: Apply the Power Rule

To simplify the expression, we will first apply the power rule, which states that for any numbers $a$ and $b$ and any integer $n$ , $(ab)^n = a^nb^n$ . We will also use the rule for negative exponents, which states that $a^{-n} = \frac{1}{a^n}$ .

Applying the Power Rule

Using the power rule, we can rewrite the expression as:

$\frac{x^{6 \cdot 1 / 3} y^{-9 \cdot 1 / 3}}{x^{-3 \cdot 1 / 3} y^{6 \cdot 1 / 3}}$

Simplifying the Exponents

Now, let's simplify the exponents:

$x^{6 \cdot 1 / 3} = x^2$ $y^{-9 \cdot 1 / 3} = y^{-3}$ $x^{-3 \cdot 1 / 3} = x^{-1}$ $y^{6 \cdot 1 / 3} = y^2$

Rewriting the Expression

Substituting the simplified exponents back into the expression, we get:

$\frac{x^2 y^{-3}}{x^{-1} y^2}$

Eliminating Negative Exponents

To eliminate the negative exponent, we will multiply the numerator and denominator by the reciprocal of the base raised to the power of the negative exponent. In this case, we will multiply by $y^3$ :

$\frac{x^2 y^{-3} \cdot y^3}{x^{-1} y^2 \cdot y^3}$

Simplifying the Expression

Simplifying the expression, we get:

$\frac{x^2 y^0}{x^{-1} y^5}$

Final Simplification

Since $y^0 = 1$ , we can simplify the expression further:

$\frac{x^2}{x^{-1} y^5}$

Using the Quotient Rule

To simplify the expression, we will use the quotient rule, which states that for any numbers $a$ and $b$ , $\frac{a^m}{a^n} = a^{m-n}$ . We will also use the rule for negative exponents, which states that $a^{-n} = \frac{1}{a^n}$ .

Applying the Quotient Rule

Using the quotient rule, we can rewrite the expression as:

$x^{2-(-1)} \cdot \frac{1}{y^5}$

Simplifying the Exponents

Simplifying the exponents, we get:

$x^3 \cdot \frac{1}{y^5}$

Final Answer

The simplified expression is:

$x^3 \cdot \frac{1}{y^5}$

This is the final answer without negative exponents.

Conclusion

Simplifying exponential expressions can be a challenging task, but with the right rules and techniques, it can be done. In this article, we have shown how to simplify an expression that contains negative exponents using the power rule, the quotient rule, and the rule for negative exponents. By following these steps, you can simplify any exponential expression and eliminate negative exponents.

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Q: What is the power rule in simplifying exponential expressions?

A: The power rule states that for any numbers $a$ and $b$ and any integer $n$ , $(ab)^n = a^nb^n$ . This rule allows us to simplify expressions by distributing the exponent to each base.

Q: How do I apply the power rule to simplify an expression?

A: To apply the power rule, simply multiply each base by the exponent. For example, if we have the expression $(xy)^3$ , we can apply the power rule by multiplying each base by 3: $x^3y^3$ .

Q: What is the quotient rule in simplifying exponential expressions?

A: The quotient rule states that for any numbers $a$ and $b$ , $\frac{a^m}{a^n} = a^{m-n}$ . This rule allows us to simplify expressions by subtracting the exponents.

Q: How do I apply the quotient rule to simplify an expression?

A: To apply the quotient rule, simply subtract the exponents. For example, if we have the expression $\frac{x^5}{x^3}$ , we can apply the quotient rule by subtracting the exponents: $x^{5-3} = x^2$ .

Q: What is the rule for negative exponents in simplifying exponential expressions?

A: The rule for negative exponents states that $a^{-n} = \frac{1}{a^n}$ . This rule allows us to rewrite negative exponents as fractions.

Q: How do I apply the rule for negative exponents to simplify an expression?

A: To apply the rule for negative exponents, simply rewrite the negative exponent as a fraction. For example, if we have the expression $x^{-3}$ , we can apply the rule by rewriting it as $\frac{1}{x^3}$ .

Q: Can I simplify an expression with multiple bases and exponents?

A: Yes, you can simplify an expression with multiple bases and exponents by applying the power rule and the quotient rule. For example, if we have the expression $(x^2y^3)^4$ , we can apply the power rule by multiplying each base by 4: $x^8y^{12}$ .

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

A: To simplify an expression with a negative exponent in the denominator, you can apply the rule for negative exponents by rewriting the negative exponent as a fraction. For example, if we have the expression $\frac{1}{x^{-3}}$ , we can apply the rule by rewriting it as $x^3$ .

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 by applying the power rule and the quotient rule. For example, if we have the expression $(x^a)^b$ , we can apply the power rule by multiplying each base by $b$ : $x^{ab}$ .

Q: How do I know when to apply the power rule and the quotient rule?

A: To determine when to apply the power rule and the quotient rule, simply look for the bases and exponents in the expression. If you see multiple bases and exponents, you can apply the power rule. If you see a fraction with exponents, you can apply the quotient rule.

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

A: Yes, you can simplify an expression with a negative exponent in the numerator by applying the rule for negative exponents. For example, if we have the expression $x^{-3}$ , we can apply the rule by rewriting it as $\frac{1}{x^3}$ .

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

A: To simplify an expression with multiple negative exponents, you can apply the rule for negative exponents by rewriting each negative exponent as a fraction. For example, if we have the expression $x^{-3}y^{-2}$ , we can apply the rule by rewriting it as $\frac{1}{x^3y^2}$ .

Q: Can I simplify an expression with a variable in the numerator and a negative exponent in the denominator?

A: Yes, you can simplify an expression with a variable in the numerator and a negative exponent in the denominator by applying the rule for negative exponents and the quotient rule. For example, if we have the expression $\frac{x^3}{x^{-2}}$ , we can apply the rule by rewriting the negative exponent as a fraction and then applying the quotient rule: $x^{3-(-2)} = x^5$ .

Q: How do I know when to use the power rule and the quotient rule together?

A: To determine when to use the power rule and the quotient rule together, simply look for the bases and exponents in the expression. If you see multiple bases and exponents, you can apply the power rule. If you see a fraction with exponents, you can apply the quotient rule. If you see both, you can apply both rules together.

Q: Can I simplify an expression with a negative exponent in the numerator and a variable in the denominator?

A: Yes, you can simplify an expression with a negative exponent in the numerator and a variable in the denominator by applying the rule for negative exponents and the quotient rule. For example, if we have the expression $\frac{x^{-3}}{y^2}$ , we can apply the rule by rewriting the negative exponent as a fraction and then applying the quotient rule: $\frac{1}{x^3y^2}$ .

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

A: To simplify an expression with multiple variables and exponents, you can apply the power rule and the quotient rule. For example, if we have the expression $(x^2y^3)^4$ , we can apply the power rule by multiplying each base by 4: $x^8y^{12}$ .

Q: Can I simplify an expression with a variable in the numerator and a negative exponent in the denominator, and multiple bases and exponents?

A: Yes, you can simplify an expression with a variable in the numerator and a negative exponent in the denominator, and multiple bases and exponents by applying the rule for negative exponents, the quotient rule, and the power rule. For example, if we have the expression $\frac{x^3y^4}{x^{-2}y^{-3}}$ , we can apply the rule by rewriting the negative exponents as fractions, then applying the quotient rule, and finally applying the power rule: $x^{3-(-2)}y^{4-(-3)} = x^5y^7$ .

Q: How do I know when to use the power rule, the quotient rule, and the rule for negative exponents together?

A: To determine when to use the power rule, the quotient rule, and the rule for negative exponents together, simply look for the bases and exponents in the expression. If you see multiple bases and exponents, you can apply the power rule. If you see a fraction with exponents, you can apply the quotient rule. If you see a negative exponent, you can apply the rule for negative exponents. If you see all three, you can apply all three rules together.