Solve (10×3)/(3×9) Raised to Power -4: Complex Fraction Expression

Q: Why do both numerator and denominator get negative exponents?

When you raise a fraction to a negative power, the entire expression gets the negative exponent. So $ \left(\frac{a×b}{c×d}\right)^{-4} = \frac{(a×b)^{-4}}{(c×d)^{-4}} $, which becomes $ \frac{a^{-4}×b^{-4}}{c^{-4}×d^{-4}} $.

Q: Can I simplify the fraction first before applying the exponent?

Yes! You could simplify $ \frac{10×3}{3×9} = \frac{10}{9} $ first, then apply $ \left(\frac{10}{9}\right)^{-4} $. However, the question asks for the corresponding expression format, so keep the original structure.

Q: What's the difference between the first and second answer choices?

Choice 1 has $ \frac{10^{-4}×3^{-4}}{3^4×9^4} $ (mixed positive/negative), while Choice 2 has $ \frac{10^{-4}×3^{-4}}{3^{-4}×9^{-4}} $ (all negative). The negative exponent applies uniformly to the entire fraction!

Q: How do I remember when exponents are positive or negative?

Think of it this way: when you have $ (\text{anything})^{-4} $, the -4 distributes to every factor inside. Don't mix and match - if the main exponent is negative, all distributed exponents stay negative.

Q: Is there a shortcut to avoid confusion with negative exponents?

Yes! Remember that $ \left(\frac{a}{b}\right)^{-n} = \left(\frac{b}{a}\right)^{n} $. But for expression matching problems like this, stick to distributing the exponent as given without flipping the fraction.

Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Simplify the following problem

00:03 According to the laws of exponents, a fraction raised to a power (N)

00:08 equals the numerator and denominator raised to the same power (N)

00:12 We will apply this formula to our exercise

00:17 We'll raise both the numerator and the denominator to the power (N)

00:24 According to the laws of exponents when the entire product is raised to a power (N)

00:28 it is equal to each factor in the product separately raised to the same power (N)

00:33 We will apply this formula to our exercise

00:46 This is the solution

Step-by-step written solution

Follow each step carefully to understand the complete solution

1

Understand the problem

Insert the corresponding expression:

$\left(\frac{10\times3}{3\times9}\right)^{-4}=$

2

Step-by-step solution

To solve this problem, we need to simplify the expression $\left(\frac{10 \times 3}{3 \times 9}\right)^{-4}$ .

Step 1: Simplify the expression inside the parentheses: $\frac{10 \times 3}{3 \times 9} = \frac{30}{27} = \frac{10}{9}$ .
Step 2: We now have $\left(\frac{10}{9}\right)^{-4}$ .
Step 3: Apply the rule for negative exponents: $\left(\frac{10}{9}\right)^{-4} = \left(\frac{9}{10}\right)^4$ .
Step 4: Apply the power of a quotient rule: $\left(\frac{9}{10}\right)^4 = \frac{9^4}{10^4}$ .
Step 5: Expand using exponentiation properties: $\frac{9^4}{10^4} = \frac{(3^2)^4}{10^4} = \frac{3^8}{10^4}$ .
Step 6: Comparing with the given choices, apply rules uniformly: distribute $4$ across numerators and denominators of the specific expression directly.
Step 7: Write using property: $\left(\frac{10 \times 3}{3 \times 9}\right)^{-4} = \frac{10^{-4}\times3^{-4}}{3^{-4}\times9^{-4}}$ .

Therefore, the correct simplified expression is $\frac{10^{-4}\times3^{-4}}{3^{-4}\times9^{-4}}$ .

3

Final Answer

$\frac{10^{-4}\times3^{-4}}{3^{-4}\times9^{-4}}$

Key Points to Remember

Essential concepts to master this topic

Rule: Apply negative exponent to entire fraction structure uniformly
Technique: Distribute exponent -4 to each factor: $(10×3)^{-4} = 10^{-4}×3^{-4}$
Check: Verify negative exponent rule applies to both numerator and denominator ✓

Common Mistakes

Avoid these frequent errors

Applying negative exponent incorrectly to individual terms
Don't convert $\left(\frac{10×3}{3×9}\right)^{-4}$ to $\frac{10^{-4}×3^{-4}}{3^4×9^4}$ = mixing positive and negative exponents! This ignores the fact that the negative exponent applies to the entire fraction. Always apply the negative exponent uniformly to both numerator and denominator factors.

Practice Quiz

Test your knowledge with interactive questions

$112^0=\text{?}$

FAQ

Everything you need to know about this question

Why do both numerator and denominator get negative exponents?

+

When you raise a fraction to a negative power, the entire expression gets the negative exponent. So $\left(\frac{a×b}{c×d}\right)^{-4} = \frac{(a×b)^{-4}}{(c×d)^{-4}}$ , which becomes $\frac{a^{-4}×b^{-4}}{c^{-4}×d^{-4}}$ .

Can I simplify the fraction first before applying the exponent?

+

Yes! You could simplify $\frac{10×3}{3×9} = \frac{10}{9}$ first, then apply $\left(\frac{10}{9}\right)^{-4}$ . However, the question asks for the corresponding expression format, so keep the original structure.

What's the difference between the first and second answer choices?

+

Choice 1 has $\frac{10^{-4}×3^{-4}}{3^4×9^4}$ (mixed positive/negative), while Choice 2 has $\frac{10^{-4}×3^{-4}}{3^{-4}×9^{-4}}$ (all negative). The negative exponent applies uniformly to the entire fraction!

How do I remember when exponents are positive or negative?

+

Think of it this way: when you have $(\text{anything})^{-4}$ , the -4 distributes to every factor inside. Don't mix and match - if the main exponent is negative, all distributed exponents stay negative.

Is there a shortcut to avoid confusion with negative exponents?

+

Yes! Remember that $\left(\frac{a}{b}\right)^{-n} = \left(\frac{b}{a}\right)^{n}$ . But for expression matching problems like this, stick to distributing the exponent as given without flipping the fraction.

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Solve (10×3)/(3×9) Raised to Power -4: Complex Fraction Expression

Negative Exponents with Complex Fractions

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