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

Q: Why does $ (-5)^3 $ equal $ -125 $ and not $ 125 $ ?

When you raise a negative number to an odd power, the result stays negative. $ (-5)^3 = (-5) \times (-5) \times (-5) = -125 $. Think of it as multiplying three negatives together!

Q: What's the difference between $ -5^{-3} $ and $ (-5)^{-3} $ ?

The parentheses matter! $ (-5)^{-3} $ means the entire negative five is raised to the power, while $ -5^{-3} $ means only 5 is raised to the power, then made negative: $ -\frac{1}{5^3} = -\frac{1}{125} $. In this case, both give the same answer!

Q: How do I remember the negative exponent rule?

Think "flip it!" A negative exponent means take the reciprocal and make the exponent positive. $ a^{-n} = \frac{1}{a^n} $. The base flips from numerator to denominator!

Q: Why isn't the answer $ \frac{1}{125} $ ?

Because $ (-5)^3 = -125 $, not $ 125 $. So $ \frac{1}{(-5)^3} = \frac{1}{-125} = -\frac{1}{125} $. The negative sign in the base affects the final answer!

Q: Can I solve this problem a different way?

Yes! You can think of it as $ (-5)^{-3} = \frac{1}{(-5)^3} = \frac{1}{(-1 \cdot 5)^3} = \frac{1}{(-1)^3 \cdot 5^3} = \frac{1}{-1 \cdot 125} = -\frac{1}{125} $. This separates the negative sign to make the calculation clearer.

Negative Exponents with Negative Base

$(-5)^{-3}=\text{?}$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Solve the following problem

00:02 According to the laws of exponents, a number (A) when raised to the power of (-N)

00:05 equals 1 divided by the number (A) raised to the power of (N)

00:08 Let's apply this to the question, the formula works from number to fraction and vice versa

00:11 We obtain 1 divided by (-5) raised to the power of (3)

00:14 According to the laws of exponents, the number (A*B) raised to the power of (N)

00:17 equals (A) raised to the power of (N) multiplied by (B) raised to the power of (N)

00:20 Let's apply this to the question

00:23 Break down (-5) into factors (-1) and (5)

00:30 We obtain (-1) to the power of 3 multiplied by (5) raised to the power of 3

00:36 Let's solve (-1) raised to the power of 3 according to the laws of exponents

00:51 Let's solve 5 raised to the power of 3 according to the laws of exponents

01:00 This is the solution

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

$(-5)^{-3}=\text{?}$

Step-by-step solution

First let's recall the negative exponent rule:

$b^{-n}=\frac{1}{b^n}$ We'll apply it to the expression we received:

$(-5)^{-3}=\frac{1}{(-5)^3}$ Next let's recall the power rule for expressions in parentheses:

$(x\cdot y)^n=x^n\cdot y^n$ And we'll apply it to the denominator of the expression we received:

$\frac{1}{(-5)^3}=\frac{1}{(-1\cdot5)^3}=\frac{1}{(-1)^3\cdot5^3}=\frac{1}{-1\cdot5^3}=-\frac{1}{5^3}=-\frac{1}{125}$ In the first step, we expressed the negative number inside the parentheses in the denominator as a multiplication between a positive number and negative one, and then we used the power rule for expressions in parentheses to expand the parentheses, and then we simplified the expression.

Let's summarize the solution to the problem:

$(-5)^{-3}=\frac{1}{(-5)^3} =\frac{1}{-5^3}=-\frac{1}{125}$

Therefore, the correct answer is answer B.

Final Answer

$-\frac{1}{125}$

Key Points to Remember

Essential concepts to master this topic

Rule: Use $b^{-n} = \frac{1}{b^n}$ to convert negative exponents
Technique: $(-5)^{-3} = \frac{1}{(-5)^3} = \frac{1}{-125} = -\frac{1}{125}$
Check: Verify $(-5)^3 = -125$ , so reciprocal gives $-\frac{1}{125}$ ✓

Common Mistakes

Avoid these frequent errors

Ignoring the negative sign when calculating the base raised to a positive power
Don't calculate $(-5)^3$ as $5^3 = 125$ = $\frac{1}{125}$ ! This ignores that $(-5)^3 = -125$ because we multiply three negative fives together. Always remember odd exponents keep the negative sign.

Practice Quiz

Test your knowledge with interactive questions

$112^0=\text{?}$

FAQ

Everything you need to know about this question

Why does $(-5)^3$ equal $-125$ and not $125$ ?

When you raise a negative number to an odd power, the result stays negative. $(-5)^3 = (-5) \times (-5) \times (-5) = -125$ . Think of it as multiplying three negatives together!

What's the difference between $-5^{-3}$ and $(-5)^{-3}$ ?

The parentheses matter! $(-5)^{-3}$ means the entire negative five is raised to the power, while $-5^{-3}$ means only 5 is raised to the power, then made negative: $-\frac{1}{5^3} = -\frac{1}{125}$ . In this case, both give the same answer!

How do I remember the negative exponent rule?

Think "flip it!" A negative exponent means take the reciprocal and make the exponent positive. $a^{-n} = \frac{1}{a^n}$ . The base flips from numerator to denominator!

Why isn't the answer $\frac{1}{125}$ ?

Because $(-5)^3 = -125$ , not $125$ . So $\frac{1}{(-5)^3} = \frac{1}{-125} = -\frac{1}{125}$ . The negative sign in the base affects the final answer!

Can I solve this problem a different way?

Yes! You can think of it as $(-5)^{-3} = \frac{1}{(-5)^3} = \frac{1}{(-1 \cdot 5)^3} = \frac{1}{(-1)^3 \cdot 5^3} = \frac{1}{-1 \cdot 125} = -\frac{1}{125}$ . This separates the negative sign to make the calculation clearer.

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Calculate (-5)^(-3): Solving Negative Base 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 does $(-5)^3$ equal $-125$ and not $125$ ?

What's the difference between $-5^{-3}$ and $(-5)^{-3}$ ?

How do I remember the negative exponent rule?

Why isn't the answer $\frac{1}{125}$ ?

Can I solve this problem a different way?

🌟 Unlock Your Math Potential

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❤️ 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 (−5)3 (-5)^3 (−5)3 equal −125 -125 −125 and not 125 125 125?

What's the difference between −5−3 -5^{-3} −5−3 and (−5)−3 (-5)^{-3} (−5)−3?

How do I remember the negative exponent rule?

Why isn't the answer 1125 \frac{1}{125} 1251​?

Can I solve this problem a different way?

🌟 Unlock Your Math Potential

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Why does $(-5)^3$ equal $-125$ and not $125$ ?

What's the difference between $-5^{-3}$ and $(-5)^{-3}$ ?

Why isn't the answer $\frac{1}{125}$ ?