Solve (-7)^(-3): Calculating Negative Number with Negative Exponent

Q: Why is the answer negative when we have a negative exponent?

The negative exponent just means "take the reciprocal" - it doesn't affect the sign! The negative comes from the base (-7) raised to an odd power. Since (-7)³ = -343, our answer $ \frac{1}{(-7)^3} = -\frac{1}{343} $ is negative.

Q: What's the difference between (-7)^{-3} and -7^{-3}?

Parentheses matter! (-7)^{-3} means the entire negative seven is the base. But -7^{-3} means -(7^{-3}), where only 7 is the base and we apply the negative sign after. They give different results!

Q: How do I remember when the answer will be positive or negative?

Focus on the base and the regular exponent! If the base is negative and the exponent is odd, the result is negative. If the exponent is even, the result is positive. The negative exponent just puts it in the denominator.

Q: Can I simplify this further than -1/343?

No, $ -\frac{1}{343} $ is already in simplest form! Since 343 = 7³, and 7 is prime, there are no common factors to cancel. You could write it as a decimal (-0.00291...), but the fraction form is usually preferred.

Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:04 Let's solve this math problem together.

00:07 Remember, when any number 'A' is raised to the power of 'N',

00:11 it equals one divided by 'A' raised to the power of negative 'N'.

00:17 Now, let's apply this rule to our question.

00:20 The number negative seven becomes one divided by negative seven,

00:26 and the exponent of negative three becomes positive three.

00:30 A negative times a negative gives us a positive.

00:35 And that's the solution to this problem!

Step-by-step written solution

Follow each step carefully to understand the complete solution

1

Understand the problem

$(-7)^{-3}=\text{?}$

2

Step-by-step solution

We begin by using the power property for a negative exponent:

$b^{-n}=\frac{1}{b^n}$ We apply it to the problem:

$(-7)^{-3}=\frac{1}{(-7)^3}$ We then subsequently notice that each whole number inside the parentheses is raised to a negative power (that is, the number and its negative coefficient together) When using the previously mentioned power property: We are careful to take this into account,

We then continue by simplifying the expression in the denominator of the fraction, remembering the exponentiation property for the power of terms in multiplication:

$(a^m)^n=a^{m\cdot n}$ We apply the resulting expression

$\frac{1}{(-7)^3}=\frac{1}{(-1\cdot7)^3}=\frac{1}{(-1)^3\cdot7^3}=\frac{1}{-1\cdot7^3}=\frac{1}{-7^3}=-\frac{1}{7^3}$

In summary we are able to deduce that the solution to the problem is as follows:

$(-7)^{-3}=\frac{1}{(-7)^3}=\frac{1}{-7^3}=-\frac{1}{7^3}$

Therefore, the correct answer is option B.

3

Final Answer

$-\frac{1}{7^{3}}$

Key Points to Remember

Essential concepts to master this topic

Rule: Negative exponent means reciprocal: $b^{-n} = \frac{1}{b^n}$
Technique: First apply exponent rule, then evaluate: $(-7)^{-3} = \frac{1}{(-7)^3}$
Check: Odd exponent keeps negative sign: $(-7)^3 = -343$ , so answer is $-\frac{1}{343}$ ✓

Common Mistakes

Avoid these frequent errors

Dropping the negative sign when applying negative exponent rule
Don't change (-7)^{-3} to 1/7^3 = ignores the negative base! This gives a positive result instead of negative. Always keep the original base intact when applying the negative exponent rule: (-7)^{-3} = 1/(-7)^3.

Practice Quiz

Test your knowledge with interactive questions

$$ (2^3)^6 = $$

FAQ

Everything you need to know about this question

Why is the answer negative when we have a negative exponent?

+

The negative exponent just means "take the reciprocal" - it doesn't affect the sign! The negative comes from the base (-7) raised to an odd power. Since (-7)³ = -343, our answer $\frac{1}{(-7)^3} = -\frac{1}{343}$ is negative.

What's the difference between (-7)^{-3} and -7^{-3}?

+

Parentheses matter! (-7)^{-3} means the entire negative seven is the base. But -7^{-3} means -(7^{-3}), where only 7 is the base and we apply the negative sign after. They give different results!

How do I remember when the answer will be positive or negative?

+

Focus on the base and the regular exponent! If the base is negative and the exponent is odd, the result is negative. If the exponent is even, the result is positive. The negative exponent just puts it in the denominator.

Can I simplify this further than -1/343?

+

No, $-\frac{1}{343}$ is already in simplest form! Since 343 = 7³, and 7 is prime, there are no common factors to cancel. You could write it as a decimal (-0.00291...), but the fraction form is usually preferred.

What if the exponent was positive instead?

+

If we had (-7)³ instead of (-7)^{-3}, we'd get -343 directly. The negative exponent is what creates the fraction. Remember: negative exponent = reciprocal of the positive exponent version!

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Solve (-7)^(-3): Calculating Negative Number with Negative Exponent

Negative Exponents with Negative Base

❤️ 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 is the answer negative when we have a negative exponent?

What's the difference between (-7)^{-3} and -7^{-3}?

How do I remember when the answer will be positive or negative?

Can I simplify this further than -1/343?

What if the exponent was positive instead?

🌟 Unlock Your Math Potential

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