Simplify the Expression: 11^-2 × 11^-5 × 11^-4 Using Exponent Laws

Q: Why do I add negative exponents instead of subtracting them?

When you see $ 11^{-2} \times 11^{-5} $, you're adding the exponents: -2 + (-5). Adding a negative number is the same as subtracting, so -2 + (-5) = -2 - 5 = -7.

Q: How do I convert a negative exponent to a positive one?

Use the rule $ a^{-n} = \frac{1}{a^n} $. So $ 11^{-11} $ becomes $ \frac{1}{11^{11}} $. The negative sign flips the base to the denominator.

Q: What's the difference between 11^11 and 1/11^11?

$ 11^{11} $ is a huge positive number, while $ \frac{1}{11^{11}} $ is a tiny positive fraction close to zero. They're completely different values!

Q: Can I use a calculator to check my answer?

Yes! Calculate $ 11^{-2} \times 11^{-5} \times 11^{-4} $ directly, then compare with $ \frac{1}{11^{11}} $. Both should give the same very small decimal result.

Q: Why can't the answer be 11^11?

Because we're multiplying negative exponents! Negative exponents create fractions, not large whole numbers. $ 11^{11} $ would be the answer if all exponents were positive.

Exponent Laws with Negative Powers

Reduce the following equation:

$11^{-2}\times11^{-5}\times11^{-4}=$

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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, the multiplication of powers with equal bases (A)

00:07 equals the same base raised to the sum of the exponents (N+M)

00:10 We'll apply this formula to our exercise

00:15 We'll maintain the base and add the exponents together

00:19 Note that we're adding a negative factor, be careful with the parentheses

00:31 A positive x A negative always equals a negative, therefore we subtract as follows

00:42 According to the laws of exponents, any number with a negative exponent (-N)

00:45 equals the reciprocal number raised to the opposite exponent (N)

00:48 We'll apply this formula to our exercise

00:51 We'll convert to the reciprocal number and raise it to the opposite exponent

00:55 This is the solution

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Reduce the following equation:

$11^{-2}\times11^{-5}\times11^{-4}=$

Step-by-step solution

To solve the expression $11^{-2} \times 11^{-5} \times 11^{-4}$ , we apply the rules for multiplying numbers with the same base:

Step 1: Use the rule for multiplying powers with the same base: $a^m \times a^n = a^{m+n}$ .
Step 2: Add the exponents: $-2 + -5 + -4$ .
Step 3: Perform the calculation: $-2 - 5 - 4 = -11$ .
Step 4: Write the expression with the combined exponent: $11^{-11}$ .
Step 5: Express $11^{-11}$ as a positive power using the property of negative exponents: $a^{-n} = \frac{1}{a^n}$ .

Therefore, $11^{-11} = \frac{1}{11^{11}}$ .

The final answer is $\frac{1}{11^{11}}$ .

Final Answer

$\frac{1}{11^{11}}$

Key Points to Remember

Essential concepts to master this topic

Rule: When multiplying same bases, add the exponents together
Technique: Calculate -2 + (-5) + (-4) = -11 step by step
Check: Convert $11^{-11}$ to $\frac{1}{11^{11}}$ using negative exponent property ✓

Common Mistakes

Avoid these frequent errors

Multiplying exponents instead of adding them
Don't multiply -2 × -5 × -4 = -40 when combining exponents! This gives $11^{-40}$ instead of the correct answer. Always add exponents when multiplying same bases: -2 + (-5) + (-4) = -11.

Practice Quiz

Test your knowledge with interactive questions

$112^0=\text{?}$

FAQ

Everything you need to know about this question

Why do I add negative exponents instead of subtracting them?

When you see $11^{-2} \times 11^{-5}$ , you're adding the exponents: -2 + (-5). Adding a negative number is the same as subtracting, so -2 + (-5) = -2 - 5 = -7.

How do I convert a negative exponent to a positive one?

Use the rule $a^{-n} = \frac{1}{a^n}$ . So $11^{-11}$ becomes $\frac{1}{11^{11}}$ . The negative sign flips the base to the denominator.

What's the difference between 11^11 and 1/11^11?

$11^{11}$ is a huge positive number, while $\frac{1}{11^{11}}$ is a tiny positive fraction close to zero. They're completely different values!

Can I use a calculator to check my answer?

Yes! Calculate $11^{-2} \times 11^{-5} \times 11^{-4}$ directly, then compare with $\frac{1}{11^{11}}$ . Both should give the same very small decimal result.

Why can't the answer be 11^11?

Because we're multiplying negative exponents! Negative exponents create fractions, not large whole numbers. $11^{11}$ would be the answer if all exponents were positive.

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