Evaluate (15/21) Raised to the Negative Third Power: Step-by-Step Solution

Q: Why does a negative exponent flip the fraction?

A negative exponent means "take the reciprocal". Since $ \left(\frac{15}{21}\right)^{-3} = \frac{1}{\left(\frac{15}{21}\right)^3} $, and dividing by a fraction means multiplying by its reciprocal, we get $ \left(\frac{21}{15}\right)^3 $!

Q: Does the negative exponent make the answer negative?

No! The negative exponent only affects the position of the base (flips it), not the sign of the answer. $ \left(\frac{15}{21}\right)^{-3} $ gives a positive result.

Q: Can I simplify the fraction first?

Yes! $ \frac{15}{21} = \frac{5}{7} $, so $ \left(\frac{15}{21}\right)^{-3} = \left(\frac{5}{7}\right)^{-3} = \left(\frac{7}{5}\right)^3 $. This makes calculations easier but doesn't change the method!

Q: What if the exponent was positive instead?

If we had $ \left(\frac{15}{21}\right)^3 $, we would keep the fraction as-is and just cube it. No flipping needed when the exponent is positive!

Q: How do I remember which way to flip?

Think: "Negative exponent = reciprocal". The reciprocal of $ \frac{15}{21} $ is $ \frac{21}{15} $. Just swap the numerator and denominator!

Negative Exponents with Fraction Bases

Insert the corresponding expression:

$\left(\frac{15}{21}\right)^{-3}=$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Simplify the following problem

00:04 According to the laws of exponents, any fraction raised to the negative exponent (-N)

00:09 equals the reciprocal fraction with the opposite exponent (N)

00:12 We will apply this formula to our exercise

00:21 This is the solution

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Insert the corresponding expression:

$\left(\frac{15}{21}\right)^{-3}=$

Step-by-step solution

To solve the expression $\left(\frac{15}{21}\right)^{-3}$ , we will apply the rule for converting negative exponents into positive exponents.

Step 1: Recognize that the negative exponent indicates the reciprocal of the base raised to the positive equivalent of the exponent. Thus, we use the formula:

$\left(\frac{a}{b}\right)^{-n} = \left(\frac{b}{a}\right)^n$ .

Step 2: Apply this formula to our expression:

$\left(\frac{15}{21}\right)^{-3} = \left(\frac{21}{15}\right)^3$ .

Therefore, the solution to the problem is $\left(\frac{21}{15}\right)^3$ , which corresponds to choice 3.

Final Answer

$\left(\frac{21}{15}\right)^3$

Key Points to Remember

Essential concepts to master this topic

Rule: Negative exponent means flip the base and make exponent positive
Technique: $\left(\frac{a}{b}\right)^{-n} = \left(\frac{b}{a}\right)^n$ converts directly
Check: Verify that $\left(\frac{15}{21}\right)^{-3} = \left(\frac{21}{15}\right)^3$ by substitution ✓

Common Mistakes

Avoid these frequent errors

Adding a negative sign to the answer
Don't think negative exponent = negative answer! $\left(\frac{15}{21}\right)^{-3} ≠ -\left(\frac{21}{15}\right)^3$ . The negative exponent only tells you to flip the fraction and make the exponent positive. Always remember: negative exponent ≠ negative result.

Practice Quiz

Test your knowledge with interactive questions

$112^0=\text{?}$

FAQ

Everything you need to know about this question

Why does a negative exponent flip the fraction?

A negative exponent means "take the reciprocal". Since $\left(\frac{15}{21}\right)^{-3} = \frac{1}{\left(\frac{15}{21}\right)^3}$ , and dividing by a fraction means multiplying by its reciprocal, we get $\left(\frac{21}{15}\right)^3$ !

Does the negative exponent make the answer negative?

No! The negative exponent only affects the position of the base (flips it), not the sign of the answer. $\left(\frac{15}{21}\right)^{-3}$ gives a positive result.

Can I simplify the fraction first?

Yes! $\frac{15}{21} = \frac{5}{7}$ , so $\left(\frac{15}{21}\right)^{-3} = \left(\frac{5}{7}\right)^{-3} = \left(\frac{7}{5}\right)^3$ . This makes calculations easier but doesn't change the method!

What if the exponent was positive instead?

If we had $\left(\frac{15}{21}\right)^3$ , we would keep the fraction as-is and just cube it. No flipping needed when the exponent is positive!

How do I remember which way to flip?

Think: "Negative exponent = reciprocal". The reciprocal of $\frac{15}{21}$ is $\frac{21}{15}$ . Just swap the numerator and denominator!

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Evaluate (15/21) Raised to the Negative Third Power: Step-by-Step Solution

Negative Exponents with Fraction Bases

❤️ 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 flip the fraction?

Does the negative exponent make the answer negative?

Can I simplify the fraction first?

What if the exponent was positive instead?

How do I remember which way to flip?

🌟 Unlock Your Math Potential

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