Simplify the Fraction: 1/(2⁷ × 9⁷ × 5⁷) Expression

Q: Why can't I just write the reciprocal as a negative number?

Negative exponents are different from negative numbers! $ 2^{-7} $ means one divided by $ 2^7 $, not negative $ 2^7 $.

Q: Do I need to calculate the actual numerical value?

Not necessarily! The question asks for the expression form using negative exponents. $ 2^{-7} \times 9^{-7} \times 5^{-7} $ is the correct simplified form.

Q: Can I combine the negative exponents somehow?

You could write it as $ (2 \times 9 \times 5)^{-7} $, but that's not one of the answer choices. Keep each base separate with its own negative exponent as shown.

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

Regular exponent rules like $ a^m \times a^n = a^{m+n} $ work with the same base. Here we have different bases (2, 9, 5), so we apply the negative exponent rule to each one individually.

Q: How do I remember the negative exponent rule?

Think: "Flip and flip!" When you have $ \frac{1}{something} $, flip the fraction position and flip the exponent sign. $ \frac{1}{a^n} $ becomes $ a^{-n} $.

Negative Exponents with Product of Powers

Insert the corresponding expression:

$\frac{1}{2^7\times9^7\times5^7}=$

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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 Break down the fraction into 3 smaller fractions

00:12 Apply the exponent laws in order to simplify the negative exponents

00:15 Convert to the reciprocal number and raise to the power of (-1)

00:19 Apply this formula to our exercise

00:29 Raise to the power of (-1)

00:48 Convert to the reciprocal number (1 divided by the number)

01:14 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:

$\frac{1}{2^7\times9^7\times5^7}=$

Step-by-step solution

To transform the given expression into one involving negative exponents, we work as follows:

Given expression: $\frac{1}{2^7 \times 9^7 \times 5^7}$

Utilizing the negative exponent rule, $\frac{1}{a^n} = a^{-n}$ , we transform each term in the denominator:

Convert $2^7$ to $2^{-7}$
Convert $9^7$ to $9^{-7}$
Convert $5^7$ to $5^{-7}$

Thus, the expression becomes:

$2^{-7} \times 9^{-7} \times 5^{-7}$

This expression corresponds to applying the negative exponent formula to each factor from the denominator.

Therefore, the solution to the problem is $2^{-7} \times 9^{-7} \times 5^{-7}$ .

Final Answer

$2^{-7}\times9^{-7}\times5^{-7}$

Key Points to Remember

Essential concepts to master this topic

Rule: When 1 is divided by a power, use negative exponents
Technique: Apply $\frac{1}{a^n} = a^{-n}$ to each factor separately
Check: Verify $2^{-7} \times 9^{-7} \times 5^{-7} = \frac{1}{2^7 \times 9^7 \times 5^7}$ ✓

Common Mistakes

Avoid these frequent errors

Combining bases before applying negative exponents
Don't combine 2×9×5 = 90 first to get $90^{-7}$ = wrong answer! This changes the mathematical value completely. Always apply the negative exponent rule $\frac{1}{a^n} = a^{-n}$ to each individual base separately.

Practice Quiz

Test your knowledge with interactive questions

$112^0=\text{?}$

FAQ

Everything you need to know about this question

Why can't I just write the reciprocal as a negative number?

Negative exponents are different from negative numbers! $2^{-7}$ means one divided by $2^7$ , not negative $2^7$ .

Do I need to calculate the actual numerical value?

Not necessarily! The question asks for the expression form using negative exponents. $2^{-7} \times 9^{-7} \times 5^{-7}$ is the correct simplified form.

Can I combine the negative exponents somehow?

You could write it as $(2 \times 9 \times 5)^{-7}$ , but that's not one of the answer choices. Keep each base separate with its own negative exponent as shown.

What's the difference between this and regular exponent rules?

Regular exponent rules like $a^m \times a^n = a^{m+n}$ work with the same base. Here we have different bases (2, 9, 5), so we apply the negative exponent rule to each one individually.

How do I remember the negative exponent rule?

Think: "Flip and flip!" When you have $\frac{1}{something}$ , flip the fraction position and flip the exponent sign. $\frac{1}{a^n}$ becomes $a^{-n}$ .

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