What Is The Value Of $n$ In The Equation Below?${ \frac{12 N}{12 5} = 12^4 }$A. -20 B. -9 C. 9 D. 20

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Introduction

In mathematics, equations involving exponents are a fundamental concept that requires a deep understanding of the properties of exponents. One of the most important properties of exponents is the quotient of powers property, which states that when we divide two powers with the same base, we can subtract the exponents. In this article, we will explore this property and use it to solve an equation involving exponents.

The Quotient of Powers Property

The quotient of powers property is a fundamental concept in mathematics that states that when we divide two powers with the same base, we can subtract the exponents. This property can be expressed mathematically as:

aman=am−n\frac{a^m}{a^n} = a^{m-n}

where aa is the base and mm and nn are the exponents.

Solving the Equation

Now, let's use the quotient of powers property to solve the equation:

12n125=124\frac{12^n}{12^5} = 12^4

To solve this equation, we can use the quotient of powers property by subtracting the exponents. This gives us:

12n−5=12412^{n-5} = 12^4

Since the bases are the same, we can equate the exponents:

n−5=4n-5 = 4

Solving for $n$

To solve for $n$, we can add 5 to both sides of the equation:

n=4+5n = 4 + 5

n=9n = 9

Therefore, the value of $n$ in the equation is 9.

Conclusion

In this article, we used the quotient of powers property to solve an equation involving exponents. We showed that when we divide two powers with the same base, we can subtract the exponents. We then used this property to solve the equation and found that the value of $n$ is 9.

Frequently Asked Questions

  • What is the quotient of powers property? The quotient of powers property is a fundamental concept in mathematics that states that when we divide two powers with the same base, we can subtract the exponents.
  • How do we use the quotient of powers property to solve an equation? To use the quotient of powers property to solve an equation, we can subtract the exponents and then equate the resulting expression to the original equation.
  • What is the value of $n$ in the equation? The value of $n$ in the equation is 9.

Final Answer

The final answer is 9.

Introduction

In our previous article, we explored the quotient of powers property and used it to solve an equation involving exponents. In this article, we will answer some frequently asked questions about solving equations with exponents.

Q&A

Q: What is the quotient of powers property?

A: The quotient of powers property is a fundamental concept in mathematics that states that when we divide two powers with the same base, we can subtract the exponents. This property can be expressed mathematically as:

aman=am−n\frac{a^m}{a^n} = a^{m-n}

where aa is the base and mm and nn are the exponents.

Q: How do we use the quotient of powers property to solve an equation?

A: To use the quotient of powers property to solve an equation, we can subtract the exponents and then equate the resulting expression to the original equation. For example, if we have the equation:

12n125=124\frac{12^n}{12^5} = 12^4

We can use the quotient of powers property to subtract the exponents:

12n−5=12412^{n-5} = 12^4

Since the bases are the same, we can equate the exponents:

n−5=4n-5 = 4

Q: What is the value of $n$ in the equation?

A: To solve for $n$, we can add 5 to both sides of the equation:

n=4+5n = 4 + 5

n=9n = 9

Therefore, the value of $n$ in the equation is 9.

Q: Can we use the quotient of powers property to solve equations with different bases?

A: No, the quotient of powers property only applies to equations with the same base. If we have an equation with different bases, we cannot use the quotient of powers property to solve it.

Q: How do we handle negative exponents?

A: When we have a negative exponent, we can rewrite it as a positive exponent by taking the reciprocal of the base. For example, if we have the equation:

1125=12−5\frac{1}{12^5} = 12^{-5}

We can rewrite the negative exponent as a positive exponent by taking the reciprocal of the base:

12−5=112512^{-5} = \frac{1}{12^5}

Q: Can we use the quotient of powers property to solve equations with fractional exponents?

A: Yes, we can use the quotient of powers property to solve equations with fractional exponents. For example, if we have the equation:

123/2121/2=121/2\frac{12^{3/2}}{12^{1/2}} = 12^{1/2}

We can use the quotient of powers property to subtract the exponents:

123/2−1/2=121/212^{3/2 - 1/2} = 12^{1/2}

Since the bases are the same, we can equate the exponents:

3/2−1/2=1/23/2 - 1/2 = 1/2

Conclusion

In this article, we answered some frequently asked questions about solving equations with exponents. We covered topics such as the quotient of powers property, handling negative exponents, and solving equations with fractional exponents.

Final Answer

The final answer is 9.