Simplify 2^6 × 2^-3: Multiplying Powers with Different Signs

Q: Why do I add the exponents when multiplying powers?

This comes from the definition of exponents! $ 2^6 $ means 2 multiplied 6 times, and $ 2^{-3} $ means dividing by 2 three times. When you multiply them, you get 6 + (-3) = 3 total factors of 2.

Q: What does a negative exponent mean?

A negative exponent means division or taking the reciprocal. For example, $ 2^{-3} = \frac{1}{2^3} = \frac{1}{8} $. So you're multiplying $ 2^6 $ by $ \frac{1}{2^3} $.

Q: How do I add a positive and negative number?

When adding 6 + (-3), think of it as 6 - 3 = 3. Adding a negative is the same as subtracting! This is why the answer is $ 2^3 $, not $ 2^9 $.

Q: Can I check my answer a different way?

Yes! Calculate each power separately: $ 2^6 = 64 $ and $ 2^{-3} = \frac{1}{8} $. Then $ 64 \times \frac{1}{8} = 8 $, which equals $ 2^3 = 8 $ ✓

Q: What if the bases were different numbers?

The exponent addition rule only works when the bases are identical. For example, you cannot simplify $ 2^6 \times 3^{-3} $ using this rule because 2 and 3 are different bases.

Exponent Rules with Negative Powers

Simplify the following equation:

$2^6\times2^{-3}=$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Simplify the following problem

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

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

00:09 We'll apply this formula to our exercise

00:12 Note that we are adding a negative factor

00:18 A positive x A negative is always negative, therefore we subtract as follows

00:21 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:

$2^6\times2^{-3}=$

Step-by-step solution

To solve the problem of simplifying $2^6 \times 2^{-3}$ , we follow these steps:

Identify the problem involves multiplying powers with the same base, $2$ .
Use the formula $a^m \times a^n = a^{m+n}$ to combine the exponents.
Add the exponents: $6 + (-3)$ .

Applying the exponent rule, we calculate:

Step 1: Given expression is $2^6 \times 2^{-3}$ .

Step 2: According to the property of exponents, add the exponents: $6 + (-3)$ .

Step 3: Simplify the exponent: $6 - 3 = 3$ .

Thus, $2^{6-3}$ .

Final Answer

$2^{6-3}$

Key Points to Remember

Essential concepts to master this topic

Rule: When multiplying powers with same base, add the exponents
Technique: For $2^6 \times 2^{-3}$ , calculate 6 + (-3) = 3
Check: Verify $2^3 = 8$ matches direct calculation ✓

Common Mistakes

Avoid these frequent errors

Multiplying exponents instead of adding
Don't multiply 6 × (-3) = -18 to get $2^{-18}$ ! This confuses multiplication of powers with raising a power to another power. Always add exponents when multiplying same bases: 6 + (-3) = 3.

Practice Quiz

Test your knowledge with interactive questions

$(3\times4\times5)^4=$

FAQ

Everything you need to know about this question

Why do I add the exponents when multiplying powers?

This comes from the definition of exponents! $2^6$ means 2 multiplied 6 times, and $2^{-3}$ means dividing by 2 three times. When you multiply them, you get 6 + (-3) = 3 total factors of 2.

What does a negative exponent mean?

A negative exponent means division or taking the reciprocal. For example, $2^{-3} = \frac{1}{2^3} = \frac{1}{8}$ . So you're multiplying $2^6$ by $\frac{1}{2^3}$ .

How do I add a positive and negative number?

When adding 6 + (-3), think of it as 6 - 3 = 3. Adding a negative is the same as subtracting! This is why the answer is $2^3$ , not $2^9$ .

Can I check my answer a different way?

Yes! Calculate each power separately: $2^6 = 64$ and $2^{-3} = \frac{1}{8}$ . Then $64 \times \frac{1}{8} = 8$ , which equals $2^3 = 8$ ✓

What if the bases were different numbers?

The exponent addition rule only works when the bases are identical. For example, you cannot simplify $2^6 \times 3^{-3}$ using this rule because 2 and 3 are different bases.

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