Which Expression Is Equivalent To 28 P 9 Q − 5 12 P − 6 Q 7 \frac{28 P^9 Q^{-5}}{12 P^{-6} Q^7} 12 P − 6 Q 7 28 P 9 Q − 5 ​ ? Assume P ≠ 0 , Q ≠ 0 P \neq 0, Q \neq 0 P  = 0 , Q  = 0 .A. 2 P 15 Q 12 \frac{2}{p^{15} Q^{12}} P 15 Q 12 2 ​ B. 7 P 15 3 Q 12 \frac{7 P^{15}}{3 Q^{12}} 3 Q 12 7 P 15 ​ C. 2 Q 12 P 15 \frac{2 Q^{12}}{p^{15}} P 15 2 Q 12 ​ D.

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Introduction

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 the process of simplifying algebraic expressions, with a focus on the given problem: 28p9q−512p−6q7\frac{28 p^9 q^{-5}}{12 p^{-6} q^7}. We will break down the solution into manageable steps, using the properties of exponents and fractions to arrive at the final answer.

Understanding Exponents

Before we dive into the solution, let's take a moment to review the properties of exponents. When we have a variable raised to a power, such as p9p^9, it means that the variable is multiplied by itself 9 times. For example, p9=p⋅p⋅p⋅p⋅p⋅p⋅p⋅p⋅pp^9 = p \cdot p \cdot p \cdot p \cdot p \cdot p \cdot p \cdot p \cdot p. When we have a negative exponent, such as q−5q^{-5}, it means that we are taking the reciprocal of the variable raised to the positive exponent. In other words, q−5=1q5q^{-5} = \frac{1}{q^5}.

Simplifying the Expression

Now that we have a solid understanding of exponents, let's tackle the given expression: 28p9q−512p−6q7\frac{28 p^9 q^{-5}}{12 p^{-6} q^7}. To simplify this expression, we will use the properties of fractions and exponents. We can start by rewriting the expression as:

28p9q−512p−6q7=2812⋅p9p−6⋅q−5q7\frac{28 p^9 q^{-5}}{12 p^{-6} q^7} = \frac{28}{12} \cdot \frac{p^9}{p^{-6}} \cdot \frac{q^{-5}}{q^7}

Using the Quotient Rule for Exponents

The next step is to apply the quotient rule for exponents, which states that when we divide two variables with the same base, we subtract the exponents. In this case, we have:

p9p−6=p9−(−6)=p9+6=p15\frac{p^9}{p^{-6}} = p^{9-(-6)} = p^{9+6} = p^{15}

Using the Product Rule for Exponents

Next, we will use the product rule for exponents, which states that when we multiply two variables with the same base, we add the exponents. In this case, we have:

q−5q7=q−5−7=q−12\frac{q^{-5}}{q^7} = q^{-5-7} = q^{-12}

Combining the Results

Now that we have simplified the expression using the quotient and product rules for exponents, we can combine the results:

28p9q−512p−6q7=2812⋅p15⋅q−12\frac{28 p^9 q^{-5}}{12 p^{-6} q^7} = \frac{28}{12} \cdot p^{15} \cdot q^{-12}

Simplifying the Fraction

The final step is to simplify the fraction 2812\frac{28}{12}. We can do this by dividing both the numerator and the denominator by their greatest common divisor, which is 4:

2812=73\frac{28}{12} = \frac{7}{3}

The Final Answer

Now that we have simplified the expression, we can write the final answer:

28p9q−512p−6q7=73p15q−12\frac{28 p^9 q^{-5}}{12 p^{-6} q^7} = \frac{7}{3} p^{15} q^{-12}

Conclusion

In this article, we have walked through the process of simplifying an algebraic expression using the properties of exponents and fractions. We have applied the quotient and product rules for exponents, and simplified the resulting expression. The final answer is 73p15q−12\frac{7}{3} p^{15} q^{-12}.

Comparison with Answer Choices

Now that we have arrived at the final answer, let's compare it with the answer choices:

  • A. 2p15q12\frac{2}{p^{15} q^{12}}
  • B. 7p153q12\frac{7 p^{15}}{3 q^{12}}
  • C. 2q12p15\frac{2 q^{12}}{p^{15}}
  • D. (no answer choice)

It's clear that the correct answer is B. 7p153q12\frac{7 p^{15}}{3 q^{12}}, which matches our final answer.

Final Thoughts

Simplifying algebraic expressions is an essential skill for any math enthusiast. By applying the properties of exponents and fractions, we can simplify even the most complex expressions. In this article, we have walked through the process of simplifying an algebraic expression, and arrived at the final answer. We hope that this article has provided a clear and concise guide to simplifying algebraic expressions.

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Q: What is the quotient rule for exponents?

A: The quotient rule for exponents states that when we divide two variables with the same base, we subtract the exponents. In other words, aman=am−n\frac{a^m}{a^n} = a^{m-n}.

Q: What is the product rule for exponents?

A: The product rule for exponents states that when we multiply two variables with the same base, we add the exponents. In other words, am⋅an=am+na^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, we can use the property that a−n=1ana^{-n} = \frac{1}{a^n}. For example, x−3=1x3x^{-3} = \frac{1}{x^3}.

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

A: Yes, you can simplify an expression with a fraction as an exponent by using the property that amn=amna^{\frac{m}{n}} = \sqrt[n]{a^m}. For example, x12=xx^{\frac{1}{2}} = \sqrt{x}.

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

A: To simplify an expression with multiple variables, you can use the properties of exponents and fractions. For example, 2x3y23x2y=23⋅x3−2⋅y2−1=23xy\frac{2x^3y^2}{3x^2y} = \frac{2}{3} \cdot x^{3-2} \cdot y^{2-1} = \frac{2}{3}xy.

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

A: Yes, you can simplify an expression with a variable in the denominator by using the property that 1an=a−n\frac{1}{a^n} = a^{-n}. For example, 1x2=x−2\frac{1}{x^2} = x^{-2}.

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

A: To simplify an expression with a coefficient, you can use the property that a⋅bm=a⋅bma \cdot b^m = a \cdot b^m. For example, 2x3=2⋅x32x^3 = 2 \cdot x^3.

Q: Can I simplify an expression with a radical?

A: Yes, you can simplify an expression with a radical by using the property that amn=amn\sqrt[n]{a^m} = a^{\frac{m}{n}}. For example, x23=x23\sqrt[3]{x^2} = x^{\frac{2}{3}}.

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

A: To simplify an expression with a negative coefficient, you can use the property that −am=−1⋅am-a^m = -1 \cdot a^m. For example, −x3=−1⋅x3-x^3 = -1 \cdot x^3.

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

A: Yes, you can simplify an expression with a variable in the numerator and denominator by using the properties of exponents and fractions. For example, x3x2=x3−2=x1=x\frac{x^3}{x^2} = x^{3-2} = x^1 = x.

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

A: To simplify an expression with a fraction as a coefficient, you can use the property that a⋅bc=a⋅bca \cdot \frac{b}{c} = \frac{a \cdot b}{c}. For example, 2⋅34=2⋅34=642 \cdot \frac{3}{4} = \frac{2 \cdot 3}{4} = \frac{6}{4}.

Q: Can I simplify an expression with a variable in the denominator and a fraction as a coefficient?

A: Yes, you can simplify an expression with a variable in the denominator and a fraction as a coefficient by using the properties of exponents and fractions. For example, 2x2⋅34=2⋅34⋅x2=64x2\frac{2}{x^2} \cdot \frac{3}{4} = \frac{2 \cdot 3}{4 \cdot x^2} = \frac{6}{4x^2}.

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

A: To simplify an expression with multiple fractions, you can use the property that ab⋅cd=a⋅cb⋅d\frac{a}{b} \cdot \frac{c}{d} = \frac{a \cdot c}{b \cdot d}. For example, 23⋅34=2⋅33⋅4=612\frac{2}{3} \cdot \frac{3}{4} = \frac{2 \cdot 3}{3 \cdot 4} = \frac{6}{12}.

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

A: Yes, you can simplify an expression with a variable in the numerator and multiple fractions in the denominator by using the properties of exponents and fractions. For example, x334⋅25=x3620=x3⋅206\frac{x^3}{\frac{3}{4} \cdot \frac{2}{5}} = \frac{x^3}{\frac{6}{20}} = \frac{x^3 \cdot 20}{6}.

Q: How do I simplify an expression with a variable in the denominator and multiple fractions in the numerator?

A: To simplify an expression with a variable in the denominator and multiple fractions in the numerator, you can use the properties of exponents and fractions. For example, 23⋅34x2=612x2=612x2\frac{\frac{2}{3} \cdot \frac{3}{4}}{x^2} = \frac{\frac{6}{12}}{x^2} = \frac{6}{12x^2}.

Q: Can I simplify an expression with a variable in the numerator and a fraction as a coefficient in the denominator?

A: Yes, you can simplify an expression with a variable in the numerator and a fraction as a coefficient in the denominator by using the properties of exponents and fractions. For example, x334=x3⋅43\frac{x^3}{\frac{3}{4}} = \frac{x^3 \cdot 4}{3}.

Q: How do I simplify an expression with a variable in the denominator and a fraction as a coefficient in the numerator?

A: To simplify an expression with a variable in the denominator and a fraction as a coefficient in the numerator, you can use the properties of exponents and fractions. For example, 23x2x3=2x23x3=23x\frac{\frac{2}{3}x^2}{x^3} = \frac{2x^2}{3x^3} = \frac{2}{3x}.

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

A: Yes, you can simplify an expression with a variable in the numerator and a variable in the denominator by using the properties of exponents and fractions. For example, x3x2=x3−2=x1=x\frac{x^3}{x^2} = x^{3-2} = x^1 = x.

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

A: To simplify an expression with a variable in the denominator and a variable in the numerator, you can use the properties of exponents and fractions. For example, x2x3=x2−3=x−1=1x\frac{x^2}{x^3} = x^{2-3} = x^{-1} = \frac{1}{x}.

Q: Can I simplify an expression with a variable in the numerator and a variable in the denominator, and a fraction as a coefficient?

A: Yes, you can simplify an expression with a variable in the numerator and a variable in the denominator, and a fraction as a coefficient by using the properties of exponents and fractions. For example, 23x2x3=2x23x3=23x\frac{\frac{2}{3}x^2}{x^3} = \frac{2x^2}{3x^3} = \frac{2}{3x}.

Q: How do I simplify an expression with a variable in the denominator and a variable in the numerator, and a fraction as a coefficient?

A: To simplify an expression with a variable in the denominator and a variable in the numerator, and a fraction as a coefficient, you can use the properties of exponents and fractions. For example, 23x3x2=2x33x2=2x3\frac{\frac{2}{3}x^3}{x^2} = \frac{2x^3}{3x^2} = \frac{2x}{3}.

Q: Can I simplify an expression with a variable in the numerator and a variable in the denominator, and multiple fractions in the numerator and denominator?

A: Yes, you can simplify an expression with a variable in the numerator and a variable in the denominator, and multiple fractions in the numerator and denominator by using the properties of exponents and fractions. For example, 23x334x2=2x3⋅43x2⋅3=8x39x2\frac{\frac{2}{3}x^3}{\frac{3}{4}x^2} = \frac{2x^3 \cdot 4}{3x^2 \cdot 3} = \frac{8x^3}{9x^2}.

Q: How do I simplify an expression with a variable in the denominator and a variable in the numerator, and multiple fractions in the numerator and denominator?

A: To simplify an expression with a variable in the denominator and a variable in the numerator, and multiple fractions in the numerator and denominator, you can use the properties of exponents and fractions. For example, $\frac{\frac{3}{4}x^2}{