Solve (1/5×6×7)^(-3): Negative Exponent with Sequential Products

Q: Why doesn't the negative exponent make the answer negative?

The negative exponent tells you to take the reciprocal, not to make the result negative! Think of it as flipping the fraction upside down, then applying the positive exponent.

Q: Do I need to calculate 5×6×7 before applying the exponent?

No! Keep it as $ (5\times6\times7)^3 $. This shows you understand the exponent rule. You could calculate $ 210^3 $ later if needed, but the expression form is the correct answer.

Q: What if I have multiple fractions with negative exponents?

Apply the same rule to each fraction separately: $ \left(\frac{1}{a}\right)^{-n} = a^n $. Each negative exponent flips its own fraction independently.

Q: How can I remember this rule easily?

Think "negative exponent = flip and positive". The negative exponent moves the base from denominator to numerator (or vice versa) and becomes positive.

Q: What's the difference between this and distributing the exponent?

Here we use the negative exponent rule first, then we have $ (5\times6\times7)^3 $. If you wanted to distribute later, it would become $ 5^3\times6^3\times7^3 $, but that's a different (more complex) form.

Negative Exponents with Fractional Bases

Insert the corresponding expression:

$\left(\frac{1}{5\times6\times7}\right)^{-3}=$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:10 Let's simplify this problem together!

00:13 First, remember the exponent rule: if we have a fraction raised to negative N, it's like flipping the fraction and using positive N.

00:22 So, the reciprocal fraction is raised to the power of N.

00:26 We'll use this idea in our example.

00:29 We'll flip our number to its reciprocal and change the exponent to positive.

00:34 And there you have it, that's the solution!

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Insert the corresponding expression:

$\left(\frac{1}{5\times6\times7}\right)^{-3}=$

Step-by-step solution

To solve the expression $\left(\frac{1}{5\times6\times7}\right)^{-3}$ , follow these steps:

Step 1: Recognize that the expression $\left(\frac{1}{5\times6\times7}\right)^{-3}$ has a negative exponent.
Step 2: Use the rule for negative exponents: $\left(\frac{1}{a}\right)^{-n} = a^{n}$ .
Step 3: Apply the rule: $\left(\frac{1}{5\times6\times7}\right)^{-3} = \left(5\times6\times7\right)^{3}$ .

Therefore, the expression simplifies to $\left(5\times6\times7\right)^3$ .

The correct answer is $\left(5\times6\times7\right)^3$ .

Final Answer

$\left(5\times6\times7\right)^3$

Key Points to Remember

Essential concepts to master this topic

Rule: Negative exponent flips fraction and makes exponent positive
Technique: $\left(\frac{1}{abc}\right)^{-n} = (abc)^n$ directly
Check: Verify $\left(\frac{1}{210}\right)^{-3} = 210^3$ by reciprocal rule ✓

Common Mistakes

Avoid these frequent errors

Applying negative sign to the result
Don't think negative exponent means negative answer = $-\left(5\times6\times7\right)^3$ ! The negative exponent only affects position (numerator vs denominator), not the sign of the result. Always remember: negative exponent means reciprocal, not negative value.

Practice Quiz

Test your knowledge with interactive questions

$(3\times4\times5)^4=$

FAQ

Everything you need to know about this question

Why doesn't the negative exponent make the answer negative?

The negative exponent tells you to take the reciprocal, not to make the result negative! Think of it as flipping the fraction upside down, then applying the positive exponent.

Do I need to calculate 5×6×7 before applying the exponent?

No! Keep it as $(5\times6\times7)^3$ . This shows you understand the exponent rule. You could calculate $210^3$ later if needed, but the expression form is the correct answer.

What if I have multiple fractions with negative exponents?

Apply the same rule to each fraction separately: $\left(\frac{1}{a}\right)^{-n} = a^n$ . Each negative exponent flips its own fraction independently.

How can I remember this rule easily?

Think "negative exponent = flip and positive". The negative exponent moves the base from denominator to numerator (or vice versa) and becomes positive.

What's the difference between this and distributing the exponent?

Here we use the negative exponent rule first, then we have $(5\times6\times7)^3$ . If you wanted to distribute later, it would become $5^3\times6^3\times7^3$ , but that's a different (more complex) form.

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