Solve (2×7×3)^(-3): Negative Exponent Expression Problem

Q: Does a negative exponent make the answer negative?

No! A negative exponent means reciprocal, not negative. $ 2^{-3} = \frac{1}{2^3} $, which is positive. The negative sign is part of the exponent rule, not the final sign.

Q: Why can't I just put a negative sign in front?

Because $ (abc)^{-3} $ means $ \frac{1}{(abc)^3} $, not $ -(abc)^3 $! The negative exponent affects each factor individually when you apply the product rule.

Q: Can I calculate the product first, then apply the exponent?

You could calculate $ 2 \times 7 \times 3 = 42 $ first, giving $ 42^{-3} $. However, the question asks for the corresponding expression using the product rule format.

Q: What's the difference between the correct and wrong answers?

The correct answer $ 2^{-3} \times 7^{-3} \times 3^{-3} $ applies the product rule properly. Wrong answers either add unnecessary negative signs or misunderstand what negative exponents mean.

Negative Exponents with Product Rule

Insert the corresponding expression:

$\left(2\times7\times3\right)^{-3}=$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Simplify the following problem

00:03 In order to open parentheses with a multiplication operation and an outside exponent

00:09 We'll raise each factor to the power

00:15 We'll apply this formula to our exercise

00:23 This is the solution

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Insert the corresponding expression:

$\left(2\times7\times3\right)^{-3}=$

Step-by-step solution

To solve the expression $(2 \times 7 \times 3)^{-3}$ , let's proceed with the following steps:

Step 1: Recognize that we have a product inside the parentheses, $2 \times 7 \times 3$ , which is raised to the power of $-3$ .

Step 2: Apply the power of a product rule, which states that $(a \times b \times c)^n = a^n \times b^n \times c^n$ . This gives us:

$(2 \times 7 \times 3)^{-3} = 2^{-3} \times 7^{-3} \times 3^{-3}$

Therefore, applying the exponent rule, we have:

$2^{-3} \times 7^{-3} \times 3^{-3}$

Final Answer

$2^{-3}\times7^{-3}\times3^{-3}$

Key Points to Remember

Essential concepts to master this topic

Product Rule: $(a \times b \times c)^n = a^n \times b^n \times c^n$
Technique: Apply exponent to each factor: $(2 \times 7 \times 3)^{-3} = 2^{-3} \times 7^{-3} \times 3^{-3}$
Check: Count factors inside parentheses equals factors with exponent applied ✓

Common Mistakes

Avoid these frequent errors

Adding negative sign to the expression
Don't write $-(2 \times 7 \times 3)^3$ or $-(2 \times 7 \times 3)^{-3}$ ! Negative exponents don't make the whole expression negative. Always apply the product rule: distribute the exponent to each factor inside the parentheses.

Practice Quiz

Test your knowledge with interactive questions

$112^0=\text{?}$

FAQ

Everything you need to know about this question

Does a negative exponent make the answer negative?

No! A negative exponent means reciprocal, not negative. $2^{-3} = \frac{1}{2^3}$ , which is positive. The negative sign is part of the exponent rule, not the final sign.

Why can't I just put a negative sign in front?

Because $(abc)^{-3}$ means $\frac{1}{(abc)^3}$ , not $-(abc)^3$ ! The negative exponent affects each factor individually when you apply the product rule.

How do I apply the product rule with negative exponents?

Take the exponent -3 and apply it to each factor separately: $2^{-3} \times 7^{-3} \times 3^{-3}$ . Each factor gets the same exponent as the original expression.

Can I calculate the product first, then apply the exponent?

You could calculate $2 \times 7 \times 3 = 42$ first, giving $42^{-3}$ . However, the question asks for the corresponding expression using the product rule format.

What's the difference between the correct and wrong answers?

The correct answer $2^{-3} \times 7^{-3} \times 3^{-3}$ applies the product rule properly. Wrong answers either add unnecessary negative signs or misunderstand what negative exponents mean.

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