Calculate 7^(-4): Evaluating Negative Integer Exponents

Q: Why does a negative exponent make a fraction?

Think of it as undoing multiplication. Since $ 7^4 $ means multiply 7 four times, $ 7^{-4} $ means divide by 7 four times, which gives you $ \frac{1}{7^4} $!

Q: Is $ 7^{-4} $ the same as $ (-7)^4 $ ?

No! $ 7^{-4} = \frac{1}{2401} $ (positive fraction), but $ (-7)^4 = 2401 $ (positive whole number). The negative sign affects the exponent, not the base.

Q: How do I calculate $ 7^4 $ quickly?

Break it down step by step: $ 7^2 = 49 $, then $ 7^4 = 49 \times 49 = 2401 $. You can also use $ 7 \times 7 \times 7 \times 7 $ if you prefer!

Q: Can negative exponents ever give negative answers?

Only if the base is negative! For example, $ (-3)^{-2} = \frac{1}{(-3)^2} = \frac{1}{9} $ (positive), but $ (-3)^{-3} = \frac{1}{(-3)^3} = \frac{1}{-27} = -\frac{1}{27} $ (negative).

Q: What's the easiest way to remember the negative exponent rule?

Remember: "Negative exponent = Flip it!" Just move the base from numerator to denominator (or vice versa) and make the exponent positive. $ 7^{-4} $ flips to $ \frac{1}{7^4} $.

Negative Exponents with Reciprocal Conversion

$7^{-4}=\text{?}$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:04 Let's solve this problem together.

00:07 Using the power laws, a number A raised to the power of negative N

00:12 is the same as one divided by A raised to the power of N.

00:17 So, let's apply this rule: it changes numbers to fractions and back.

00:23 We get one divided by seven raised to the power of four.

00:28 Now, let's solve seven raised to the power of four using these laws.

00:34 And there you go, that's the solution!

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

$7^{-4}=\text{?}$

Step-by-step solution

We must first remind ourselves of the negative exponent rule:

$a^{-n}=\frac{1}{a^n}$ When applied to given the expression we obtain the following:

$7^{-4}=\frac{1}{7^4}=\frac{1}{2401}$

Therefore, the correct answer is option C.

Final Answer

$\frac{1}{2401}$

Key Points to Remember

Essential concepts to master this topic

Rule: Negative exponents mean reciprocal: $a^{-n} = \frac{1}{a^n}$
Technique: Convert $7^{-4}$ to $\frac{1}{7^4}$ , then calculate $7^4 = 2401$
Check: Verify $7^{-4} \times 7^4 = 7^0 = 1$ using exponent rules ✓

Common Mistakes

Avoid these frequent errors

Multiplying base by negative exponent
Don't calculate 7 × (-4) = -28! Negative exponents don't mean multiply by a negative number. Always flip to reciprocal first: $7^{-4} = \frac{1}{7^4}$ , then evaluate the positive exponent.

Practice Quiz

Test your knowledge with interactive questions

Insert the corresponding expression:

$\left(\frac{1}{20}\right)^{-7}=$

FAQ

Everything you need to know about this question

Why does a negative exponent make a fraction?

Think of it as undoing multiplication. Since $7^4$ means multiply 7 four times, $7^{-4}$ means divide by 7 four times, which gives you $\frac{1}{7^4}$ !

Is $7^{-4}$ the same as $(-7)^4$ ?

No! $7^{-4} = \frac{1}{2401}$ (positive fraction), but $(-7)^4 = 2401$ (positive whole number). The negative sign affects the exponent, not the base.

How do I calculate $7^4$ quickly?

Break it down step by step: $7^2 = 49$ , then $7^4 = 49 \times 49 = 2401$ . You can also use $7 \times 7 \times 7 \times 7$ if you prefer!

Can negative exponents ever give negative answers?

Only if the base is negative! For example, $(-3)^{-2} = \frac{1}{(-3)^2} = \frac{1}{9}$ (positive), but $(-3)^{-3} = \frac{1}{(-3)^3} = \frac{1}{-27} = -\frac{1}{27}$ (negative).

What's the easiest way to remember the negative exponent rule?

Remember: "Negative exponent = Flip it!" Just move the base from numerator to denominator (or vice versa) and make the exponent positive. $7^{-4}$ flips to $\frac{1}{7^4}$ .

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Calculate 7^(-4): Evaluating Negative Integer Exponents

Negative Exponents with Reciprocal Conversion

❤️ Continue Your Math Journey!

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 does a negative exponent make a fraction?

Is $7^{-4}$ the same as $(-7)^4$ ?

How do I calculate $7^4$ quickly?

Can negative exponents ever give negative answers?

What's the easiest way to remember the negative exponent rule?

🌟 Unlock Your Math Potential

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Calculate 7^(-4): Evaluating Negative Integer Exponents

Negative Exponents with Reciprocal Conversion

❤️ Continue Your Math Journey!

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 does a negative exponent make a fraction?

Is 7−4 7^{-4} 7−4 the same as (−7)4 (-7)^4 (−7)4?

How do I calculate 74 7^4 74 quickly?

Can negative exponents ever give negative answers?

What's the easiest way to remember the negative exponent rule?

🌟 Unlock Your Math Potential

More Questions

Applying Combined Exponents Rules

Solve (3×2×4×6)^(-4): Negative Exponent Calculation

Calculate (-5)^(-3): Solving Negative Base with Negative Exponent

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Solve: (2²)³ + (3³)⁴ + (9²)⁶ - Nested Exponents Challenge

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Is $7^{-4}$ the same as $(-7)^4$ ?

How do I calculate $7^4$ quickly?