Simplify 1/((2×7×8)^9): Complex Fraction with Power Expression

Q: Why does the reciprocal become a negative exponent?

The negative exponent rule is a shorthand for reciprocals! Instead of writing the fraction $ \frac{1}{a^n} $, we can write $ a^{-n} $ - they mean exactly the same thing.

Q: Do I need to calculate 2×7×8 first?

No! Keep the expression (2×7×8) as one unit. The rule works with any base, whether it's a single number or a multiplication like this.

Q: What's the difference between negative base and negative exponent?

A negative exponent like $ x^{-3} $ means reciprocal, while a negative base like $ (-x)^3 $ affects the sign. Don't confuse them!

Q: How do I check if my answer is correct?

Take the reciprocal of your answer! If $ (2×7×8)^{-9} $ is correct, then $ \frac{1}{(2×7×8)^{-9}} $ should equal $ (2×7×8)^9 $ - the original denominator.

Q: Can I distribute the negative exponent?

No! The negative exponent applies to the entire base $ (2×7×8) $. You cannot split it into $ 2^{-9} × 7^{-9} × 8^{-9} $ until you first apply the negative exponent rule.

Negative Exponents with Reciprocal Expressions

Insert the corresponding expression:

$\frac{1}{\left(2\times7\times8\right)^9}=$

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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 Apply the exponent laws in order to simplify negative exponents

00:06 Convert to the reciprocal (1 divided by the number)

00:10 Raise it to the opposite power (N)

00:12 We'll apply this formula to our exercise

00:16 Convert to the reciprocal (1 divided by the number)

00:23 Proceed to raise it to the opposite power

00:26 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:

$\frac{1}{\left(2\times7\times8\right)^9}=$

Step-by-step solution

To solve this expression, we apply the rule for negative exponents.

The expression given is $\frac{1}{(2 \times 7 \times 8)^9}$ . We recognize this as the reciprocal of a power, which can be rewritten using the negative exponent rule:

$\frac{1}{a^b} = a^{-b}$

Thus, $\frac{1}{(2 \times 7 \times 8)^9}$ can be rewritten as $(2 \times 7 \times 8)^{-9}$ .

Thus, the correct answer is choice 1: $(2 \times 7 \times 8)^{-9}$ .

Final Answer

$\left(2\times7\times8\right)^{-9}$

Key Points to Remember

Essential concepts to master this topic

Rule: $\frac{1}{a^n} = a^{-n}$ for any non-zero base
Technique: Keep base unchanged: $\frac{1}{(2×7×8)^9} = (2×7×8)^{-9}$
Check: Reciprocal of negative exponent returns original: $\frac{1}{(abc)^{-9}} = (abc)^9$ ✓

Common Mistakes

Avoid these frequent errors

Adding negative signs to the base
Don't write -(2×7×8)^(-9) when converting reciprocals = wrong sign! The negative exponent replaces the fraction, but the base stays positive. Always keep the base exactly the same and only change the exponent sign.

Practice Quiz

Test your knowledge with interactive questions

$112^0=\text{?}$

FAQ

Everything you need to know about this question

Why does the reciprocal become a negative exponent?

The negative exponent rule is a shorthand for reciprocals! Instead of writing the fraction $\frac{1}{a^n}$ , we can write $a^{-n}$ - they mean exactly the same thing.

Do I need to calculate 2×7×8 first?

No! Keep the expression (2×7×8) as one unit. The rule works with any base, whether it's a single number or a multiplication like this.

What's the difference between negative base and negative exponent?

A negative exponent like $x^{-3}$ means reciprocal, while a negative base like $(-x)^3$ affects the sign. Don't confuse them!

How do I check if my answer is correct?

Take the reciprocal of your answer! If $(2×7×8)^{-9}$ is correct, then $\frac{1}{(2×7×8)^{-9}}$ should equal $(2×7×8)^9$ - the original denominator.

Can I distribute the negative exponent?

No! The negative exponent applies to the entire base $(2×7×8)$ . You cannot split it into $2^{-9} × 7^{-9} × 8^{-9}$ until you first apply the negative exponent rule.

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