Solve Fraction with Negative Exponent: 2/4^(-2)

Q: Why does the negative exponent flip the fraction?

A negative exponent means "take the reciprocal and make the exponent positive." So $ 4^{-2} = \frac{1}{4^2} $. When you divide by a fraction, you multiply by its reciprocal!

Q: How do I handle division by a negative exponent?

Division by a negative exponent becomes multiplication! $ \frac{2}{4^{-2}} = 2 \times \frac{1}{4^{-2}} = 2 \times 4^2 $. The negative exponent flips to positive.

Q: Can I convert to the same base like in the explanation?

Yes! Converting $ 4 = 2^2 $ is helpful for complex problems. You get $ \frac{2}{2^{-4}} = 2^{1-(-4)} = 2^5 = 32 $. Both methods give the same answer!

Q: What's the difference between 4^(-2) and -4^2?

$ 4^{-2} = \frac{1}{16} $ while $ -4^2 = -16 $. The placement of the negative sign matters! Negative exponent affects the position, negative coefficient affects the sign.

Q: How do I know when my final answer is simplified?

Your answer is simplified when you have positive exponents only and no fractions in denominators. $ 2 \cdot 4^2 = 2 \cdot 16 = 32 $ is fully simplified.

Negative Exponent Rules with Fraction Division

$\frac{2}{4^{-2}}=\text{?}$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:07 Let's simplify this expression step by step.

00:11 If you have one divided by a number with a negative exponent...

00:15 ...it becomes the same base with a positive exponent.

00:19 So, let's apply this to our problem.

00:23 Four to the power of negative exponent becomes four with a positive exponent.

00:29 Next, we'll break down four to two squared.

00:33 Remember, when there's a power of a power, multiply the exponents.

00:39 Let's see how this works in our exercise.

00:44 Now, solve the multiplication of the exponents.

00:49 When multiplying powers with the same base...

00:53 ...add the exponents together for the result's power.

00:58 Let's apply this to finish our problem.

01:02 Solve the exponent, and that's how you find the solution!

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

$\frac{2}{4^{-2}}=\text{?}$

Step-by-step solution

First, let's note that 4 is a power of 2:

$4=2^2$ therefore we can perform a conversion to a common base for all terms in the problem,

Let's apply this:

$\frac{2}{4^{-2}}=\frac{2}{(2^2)^{-2}}$ Next, we'll use the power law for power of power:

$(a^m)^n=a^{m\cdot n}$ and we'll apply this law to the denominator term we got in the last step:

$\frac{2}{(2^2)^{-2}}=\frac{2}{2^{2\cdot(-2)}}=\frac{2}{2^{-4}}$ where in the first step we applied the above law to the denominator and in the second step we simplified the expression we got,

Next, we'll use the power law for division between terms with identical bases:

$\frac{a^m}{a^n}=a^{m-n}$ and we'll apply this law to the last expression we got:

$\frac{2}{2^{-4}}=2^{1-(-4)}=2^{1+4}=2^5$

Therefore the correct answer is answer B.

Final Answer

$2\cdot4^2$

Key Points to Remember

Essential concepts to master this topic

Rule: Negative exponents flip to positive in the opposite position
Technique: Convert $4^{-2}$ to $\frac{1}{4^2}$ then simplify
Check: Final answer $2 \cdot 4^2 = 32$ matches calculation ✓

Common Mistakes

Avoid these frequent errors

Making the entire fraction negative
Don't think $\frac{2}{4^{-2}}$ becomes negative = wrong sign! The negative is only in the exponent, not affecting the overall sign. Always remember that $a^{-n} = \frac{1}{a^n}$ without changing signs.

Practice Quiz

Test your knowledge with interactive questions

$112^0=\text{?}$

FAQ

Everything you need to know about this question

Why does the negative exponent flip the fraction?

A negative exponent means "take the reciprocal and make the exponent positive." So $4^{-2} = \frac{1}{4^2}$ . When you divide by a fraction, you multiply by its reciprocal!

How do I handle division by a negative exponent?

Division by a negative exponent becomes multiplication! $\frac{2}{4^{-2}} = 2 \times \frac{1}{4^{-2}} = 2 \times 4^2$ . The negative exponent flips to positive.

Can I convert to the same base like in the explanation?

Yes! Converting $4 = 2^2$ is helpful for complex problems. You get $\frac{2}{2^{-4}} = 2^{1-(-4)} = 2^5 = 32$ . Both methods give the same answer!

What's the difference between 4^(-2) and -4^2?

$4^{-2} = \frac{1}{16}$ while $-4^2 = -16$ . The placement of the negative sign matters! Negative exponent affects the position, negative coefficient affects the sign.

How do I know when my final answer is simplified?

Your answer is simplified when you have positive exponents only and no fractions in denominators. $2 \cdot 4^2 = 2 \cdot 16 = 32$ is fully simplified.

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Solve Fraction with Negative Exponent: 2/4^(-2)

Negative Exponent Rules with Fraction Division

❤️ 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 the negative exponent flip the fraction?

How do I handle division by a negative exponent?

Can I convert to the same base like in the explanation?

What's the difference between 4^(-2) and -4^2?

How do I know when my final answer is simplified?

🌟 Unlock Your Math Potential

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