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Question

Difficulty: EasyOperations with Variable Exponents

Find the value of nn that makes the following equation true:

x9x3=xn\frac{x^9}{x^3} = x^n

Answer: 6 / six

Answer

6
According to the quotient rule for exponents, when you divide two exponential terms that have the same base, you subtract the exponent of the denominator from the exponent of the numerator. In this equation, both terms have the base xx. Subtracting the denominator's exponent (3) from the numerator's exponent (9) gives 9−3=69 - 3 = 6. Therefore, the left side simplifies to x6x^6. Equating this to xnx^n means that nn must equal 6.

Step-by-Step Solution

1
Apply the quotient rule for exponents, which states that when dividing terms with identical bases, you subtract the exponent in the denominator from the exponent in the numerator.
x9−3=xnx^{9-3} = x^n
To simplify the left side of the equation into a single term with base xx.
2
Perform the subtraction in the exponent.
x6=xnx^6 = x^n
Calculating 9−39 - 3 gives the simplified exponent.
3
Set the resulting exponent equal to nn.
n=6n = 6
Since the bases are identical (xx), their exponents must be equal for the equation to hold true.

Key Concept

Quotient Rule for Exponents
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Topics

Operations with Variable Exponents

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