Simplify Complex Expression: (x^4⋅x^8⋅x^(-3))/(x^(-4)) ÷ (x^10⋅x^3)/x^5

Exponent Rules with Variable Comparison

x4x8x3x4——x10x3x5 \frac{x^4x^8x^{-3}}{x^{-4}}_{——}\frac{x^{10}x^3}{x^5}

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations
00:00 Insert the correct sign
00:03 When multiplying powers with equal bases
00:07 The power of the result equals the sum of the powers
00:11 We'll apply this formula to our exercise and proceed to add together the powers
00:18 We'll apply the same method to the right side
00:27 Let's calculate the numerator's powers
00:35 When dividing powers with equal bases
00:38 The power of the result equals the difference of powers
00:42 We'll apply this formula to our exercise and proceed to subtract the powers
00:55 If the variable is negative, then the number will be negative
01:02 Since we don't know, we cannot determine whether this is true
01:05 This is the solution

Step-by-step written solution

Follow each step carefully to understand the complete solution
1

Understand the problem

x4x8x3x4——x10x3x5 \frac{x^4x^8x^{-3}}{x^{-4}}_{——}\frac{x^{10}x^3}{x^5}

2

Step-by-step solution

To solve the problem, follow these steps to simplify both expressions using exponent rules:

First Expression: x4x8x3x4\frac{x^4 x^8 x^{-3}}{x^{-4}}

  • Combine the exponents in the numerator: x4+83=x9x^{4+8-3} = x^9.
  • Apply the quotient rule: x9x4=x9(4)=x9+4=x13\frac{x^9}{x^{-4}} = x^{9 - (-4)} = x^{9 + 4} = x^{13}.

Second Expression: x10x3x5\frac{x^{10} x^3}{x^5}

  • Combine the exponents in the numerator: x10+3=x13x^{10+3} = x^{13}.
  • Apply the quotient rule: x13x5=x135=x8\frac{x^{13}}{x^5} = x^{13-5} = x^8.

Now we compare the results:
First Expression: x13x^{13}
Second Expression: x8x^8

In general, x13x^{13} is larger than x8x^8 for positive x x , but since xx could be any real number not zero, the full comparison could vary for negative xx.
Therefore, given the context, it's difficult to calculate a single universal answer without more information on the value of xx.

Thus, it is not possible to calculate universally between these expressions.

Therefore, our answer is Choice 4: It is not possible to calculate.

3

Final Answer

It is not possible to calculate

Key Points to Remember

Essential concepts to master this topic
  • Product Rule: When multiplying powers, add the exponents: xaxb=xa+b x^a \cdot x^b = x^{a+b}
  • Quotient Rule: When dividing powers, subtract exponents: x13x4=x13(4)=x17 \frac{x^{13}}{x^{-4}} = x^{13-(-4)} = x^{17}
  • Check Variables: Compare x13 x^{13} vs x8 x^8 depends on x value ✓

Common Mistakes

Avoid these frequent errors
  • Assuming higher exponents always mean larger values
    Don't assume x13>x8 x^{13} > x^8 for all x values = wrong conclusion! When x is negative, the comparison changes based on whether exponents are odd or even. Always consider that x can be positive, negative, or between -1 and 1.

Practice Quiz

Test your knowledge with interactive questions

\( 112^0=\text{?} \)

FAQ

Everything you need to know about this question

Why can't I just compare the exponents 13 and 8?

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Because the value of x matters! If x = 0.5, then x13<x8 x^{13} < x^8 . If x = -1, then x13=1 x^{13} = -1 but x8=1 x^8 = 1 . The relationship changes!

How do I simplify expressions with negative exponents?

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Remember that negative exponents mean reciprocals: xn=1xn x^{-n} = \frac{1}{x^n} . So 1x4=x4 \frac{1}{x^{-4}} = x^4 , making division become multiplication.

What does 'not possible to calculate' mean here?

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It means we cannot determine a universal relationship between the two expressions without knowing the specific value of x. The comparison depends entirely on what x equals!

Can I substitute a value for x to check my work?

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Yes! Try x = 2: First expression gives 213=8192 2^{13} = 8192 , second gives 28=256 2^8 = 256 . But try x = 0.5 and see how the relationship flips!

Why do I add exponents when multiplying?

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Because xaxb x^a \cdot x^b means x multiplied by itself (a+b) times total. For example: x2x3=(xx)(xxx)=x5 x^2 \cdot x^3 = (x \cdot x) \cdot (x \cdot x \cdot x) = x^5 .

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