Solve 19^-2: Calculating a Specific Negative Exponent

Q: Why isn't the answer negative if the exponent is negative?

The negative sign in the exponent doesn't make the result negative! It tells you to take the reciprocal (flip) of the positive power. So $ 19^{-2} = \frac{1}{19^2} $, which is positive.

Q: How do I calculate 19 squared quickly?

Break it down: $ 19^2 = (20-1)^2 = 20^2 - 2(20)(1) + 1^2 = 400 - 40 + 1 = 361 $. Or simply multiply: 19 × 19 = 361.

Q: What's the difference between -2^19 and 19^-2?

These are completely different! $ 19^{-2} = \frac{1}{361} $ (small positive fraction), while $ (-2)^{19} $ is a huge negative number. The position of the negative sign matters!

Q: Can I use a calculator for this?

Yes! Enter 19, then the x^y or ^ button, then -2. But understanding the rule helps you solve it by hand and catch calculator errors.

Q: Will the answer always be a unit fraction?

When the base is a whole number greater than 1 and the exponent is negative, yes! You'll get $ \frac{1}{\text{some whole number}} $, which is a unit fraction.

Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Solving an exercise with a negative exponent

00:03 According to the laws of exponents, any number(A)raised to the power of(N)

00:06 equals 1 divided by the number(A)raised to the power of (-N)

00:13 The number(19) becomes 1 divided by(19)

00:15 The exponent(-2) changes its sign to a positive -(2-)

00:16 and the exponent(-2) becomes -(-2)

00:19 A negative multiplied by a negative becomes a positive hence the exponent is 2

00:22 This is the solution

Step-by-step written solution

Follow each step carefully to understand the complete solution

1

Understand the problem

$19^{-2}=\text{?}$

2

Step-by-step solution

In order to solve the exercise, we use the negative exponent rule.

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

We apply the rule to the given exercise:

$19^{-2}=\frac{1}{19^2}$

We can then continue and calculate the exponent.

$\frac{1}{19^2}=\frac{1}{361}$

3

Final Answer

$\frac{1}{361}$

Key Points to Remember

Essential concepts to master this topic

Rule: Negative exponent means take the reciprocal of positive power
Technique: Convert $19^{-2}$ to $\frac{1}{19^2}$ then calculate $\frac{1}{361}$
Check: Multiply answer by original base squared: $\frac{1}{361} \times 361 = 1$ ✓

Common Mistakes

Avoid these frequent errors

Making the result negative because of negative exponent
Don't think $19^{-2}$ equals -361 or -38! The negative exponent doesn't make the answer negative, it creates a reciprocal. Always remember: negative exponent = flip to fraction form.

Practice Quiz

Test your knowledge with interactive questions

$112^0=\text{?}$

FAQ

Everything you need to know about this question

Why isn't the answer negative if the exponent is negative?

+

The negative sign in the exponent doesn't make the result negative! It tells you to take the reciprocal (flip) of the positive power. So $19^{-2} = \frac{1}{19^2}$ , which is positive.

How do I calculate 19 squared quickly?

+

Break it down: $19^2 = (20-1)^2 = 20^2 - 2(20)(1) + 1^2 = 400 - 40 + 1 = 361$ . Or simply multiply: 19 × 19 = 361.

What's the difference between -2^19 and 19^-2?

+

These are completely different! $19^{-2} = \frac{1}{361}$ (small positive fraction), while $(-2)^{19}$ is a huge negative number. The position of the negative sign matters!

Can I use a calculator for this?

+

Yes! Enter 19, then the x^y or ^ button, then -2. But understanding the rule helps you solve it by hand and catch calculator errors.

Will the answer always be a unit fraction?

+

When the base is a whole number greater than 1 and the exponent is negative, yes! You'll get $\frac{1}{\text{some whole number}}$ , which is a unit fraction.

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Solve 19^-2: Calculating a Specific Negative Exponent

Negative Exponents with Power Calculations

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Step-by-step video solution

Step-by-step written solution

Understand the problem

Step-by-step solution

Final Answer

Key Points to Remember

Common Mistakes

Practice Quiz

FAQ

Why isn't the answer negative if the exponent is negative?

How do I calculate 19 squared quickly?

What's the difference between -2^19 and 19^-2?

Can I use a calculator for this?

Will the answer always be a unit fraction?

🌟 Unlock Your Math Potential

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