Solve (a×b×c)^(-2): Negative Exponent with Multiple Variables

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

The negative exponent is about position, not sign! It tells you to flip the expression to the denominator. Think of $ x^{-1} = \frac{1}{x} $, not $ -x $.

Q: Do I need to expand (a×b×c)² in the denominator?

Not necessarily! You can leave it as $ \frac{1}{(abc)^2} $ or expand to $ \frac{1}{a^2b^2c^2} $. Both forms are equally correct.

Q: What if there are numbers mixed with variables?

The same rule applies! For example: $ (2ab)^{-3} = \frac{1}{(2ab)^3} = \frac{1}{8a^3b^3} $. The negative exponent affects the entire expression.

Q: Can I split the negative exponent across the variables?

Yes! $ (abc)^{-2} = a^{-2} \times b^{-2} \times c^{-2} = \frac{1}{a^2} \times \frac{1}{b^2} \times \frac{1}{c^2} = \frac{1}{a^2b^2c^2} $. Both methods give the same result!

Q: How do I remember the negative exponent rule?

Think "negative means flip!" Just like $ 2^{-1} = \frac{1}{2} $, any negative exponent flips the base to the opposite position (numerator ↔ denominator).

Negative Exponents with Product Expressions

Solve the following equation: $\left(a\times b\times c\right)^{-2}=$

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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 According to the laws of exponents, a number with a negative exponent (-N)

00:06 equals the reciprocal number raised to the same power multiplied by (-1)

00:09 We will apply this formula to our exercise

00:12 Convert to the reciprocal number

00:18 Raise it to the same power (N) multiplied by (-1)

00:22 This is the solution

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Solve the following equation: $\left(a\times b\times c\right)^{-2}=$

Step-by-step solution

To solve the equation $(a \times b \times c)^{-2}$ , we'll make use of the exponent rules. The rule for a negative exponent states that $x^{-n} = \frac{1}{x^n}$ . Therefore, applying this rule directly to our expression, we obtain:

$(a \times b \times c)^{-2} = \frac{1}{(a \times b \times c)^2}$ .

Let's break this down step-by-step:

Step 1: Recognize the need to convert the negative exponent. According to the exponent rule, any expression with a negative exponent can be rewritten as a reciprocal with a positive exponent.
Step 2: Apply the rule: $(a \times b \times c)^{-2}$ becomes $\frac{1}{(a \times b \times c)^2}$ .

By applying the rules correctly, we have simplified the expression to:

$\frac{1}{(a \times b \times c)^2}$ .

Final Answer

$\frac{1}{\left(a\times b\times c\right)^2}$

Key Points to Remember

Essential concepts to master this topic

Rule: Negative exponent means reciprocal with positive exponent
Technique: $(abc)^{-2} = \frac{1}{(abc)^2}$ using reciprocal rule
Check: Verify the denominator has positive exponent and no negative signs ✓

Common Mistakes

Avoid these frequent errors

Adding negative sign to the expression
Don't think $(abc)^{-2} = -(abc)^2$ ! The negative exponent creates a reciprocal, not a negative value. Always remember that negative exponents flip to reciprocals with positive exponents.

Practice Quiz

Test your knowledge with interactive questions

Choose the expression that corresponds to the following:

$$ $\left(2\times11\right)^5=$

FAQ

Everything you need to know about this question

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

The negative exponent is about position, not sign! It tells you to flip the expression to the denominator. Think of $x^{-1} = \frac{1}{x}$ , not $-x$ .

Do I need to expand (a×b×c)² in the denominator?

Not necessarily! You can leave it as $\frac{1}{(abc)^2}$ or expand to $\frac{1}{a^2b^2c^2}$ . Both forms are equally correct.

What if there are numbers mixed with variables?

The same rule applies! For example: $(2ab)^{-3} = \frac{1}{(2ab)^3} = \frac{1}{8a^3b^3}$ . The negative exponent affects the entire expression.

Can I split the negative exponent across the variables?

Yes! $(abc)^{-2} = a^{-2} \times b^{-2} \times c^{-2} = \frac{1}{a^2} \times \frac{1}{b^2} \times \frac{1}{c^2} = \frac{1}{a^2b^2c^2}$ . Both methods give the same result!

How do I remember the negative exponent rule?

Think "negative means flip!" Just like $2^{-1} = \frac{1}{2}$ , any negative exponent flips the base to the opposite position (numerator ↔ denominator).

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