Simplify 4^-2 × 4^-4: Negative Exponents Multiplication

Q: Why do we add exponents when multiplying powers?

Think of it this way: $ 4^3 \times 4^2 $ means (4×4×4) × (4×4), which gives us 4 multiplied 5 times total, or $ 4^5 $. The same logic applies to negative exponents!

Q: What does a negative exponent actually mean?

A negative exponent means "take the reciprocal". So $ 4^{-2} = \frac{1}{4^2} = \frac{1}{16} $. It's not a negative number - it's a positive fraction!

Q: Can I just ignore the negative signs and add 2 + 4?

No! The negative signs are crucial. Ignoring them would give you $ 4^6 $ instead of $ 4^{-6} $ - that's the difference between 4,096 and $ \frac{1}{4096} $!

Q: Why isn't the answer 4^(-4+2)?

Order matters in subtraction! $ 4^{-2} \times 4^{-4} $ means we start with the first exponent (-2) and add the second (-4). So it's -2 + (-4) = -6, not -4 + 2 = -2.

Exponent Rules with Negative Powers

Simplify the following equation:

$4^{-2}\times4^{-4}=$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Simplify the following problem

00:03 According to the laws of exponents, the multiplication of powers with an equal base (A)

00:06 equals the same base raised to the sum of the exponents (N+M)

00:09 We will apply this formula to our exercise

00:13 Note that we are adding together negative numbers

00:17 A positive x A negative is always negative, therefore we subtract as follows

00:23 This is the solution

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Simplify the following equation:

$4^{-2}\times4^{-4}=$

Step-by-step solution

To solve this problem, we'll follow these steps:

Step 1: Identify that both terms have the same base, which is 4.
Step 2: Use the exponent rule for multiplication of powers with the same base: $a^m \times a^n = a^{m+n}$ .
Step 3: Add the exponents $-2$ and $-4$ .

Now, let's work through these steps:

Step 1: We have the expression $4^{-2} \times 4^{-4}$ .

Step 2: Applying the exponent rule, we combine the exponents:

$4^{-2} \times 4^{-4} = 4^{-2 + (-4)}$

Therefore, our answer is $4^{-2-4}$ , which matches choice 4.

Final Answer

$4^{-2-4}$

Key Points to Remember

Essential concepts to master this topic

Rule: When multiplying same bases, add the exponents: $a^m \times a^n = a^{m+n}$
Technique: Add negative exponents: $4^{-2} \times 4^{-4} = 4^{-2+(-4)} = 4^{-6}$
Check: Convert to fractions: $\frac{1}{4^2} \times \frac{1}{4^4} = \frac{1}{4^6}$ ✓

Common Mistakes

Avoid these frequent errors

Multiplying the exponents instead of adding them
Don't multiply -2 × -4 = 8 to get $4^8$ ! Multiplication rule applies to powers, not exponents. This gives a completely wrong positive result instead of the correct negative exponent. Always add exponents when multiplying same bases: $4^{-2} \times 4^{-4} = 4^{-2+(-4)} = 4^{-6}$ .

Practice Quiz

Test your knowledge with interactive questions

$112^0=\text{?}$

FAQ

Everything you need to know about this question

Why do we add exponents when multiplying powers?

Think of it this way: $4^3 \times 4^2$ means (4×4×4) × (4×4), which gives us 4 multiplied 5 times total, or $4^5$ . The same logic applies to negative exponents!

What does a negative exponent actually mean?

A negative exponent means "take the reciprocal". So $4^{-2} = \frac{1}{4^2} = \frac{1}{16}$ . It's not a negative number - it's a positive fraction!

How do I add negative numbers like -2 + (-4)?

When adding negative numbers, you're moving further left on the number line. -2 + (-4) = -6. Think of it as "negative 2, plus another negative 4, gives negative 6."

Can I just ignore the negative signs and add 2 + 4?

No! The negative signs are crucial. Ignoring them would give you $4^6$ instead of $4^{-6}$ - that's the difference between 4,096 and $\frac{1}{4096}$ !

Why isn't the answer 4^(-4+2)?

Order matters in subtraction! $4^{-2} \times 4^{-4}$ means we start with the first exponent (-2) and add the second (-4). So it's -2 + (-4) = -6, not -4 + 2 = -2.

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