Simplify Powers of 2: 2³ × 2⁴ × 2⁶ × 2⁵

Q: Why do we add exponents instead of multiplying them?

The rule $ a^m \times a^n = a^{m+n} $ comes from repeated multiplication. For example, $ 2^3 \times 2^2 = (2 \times 2 \times 2) \times (2 \times 2) = 2^5 $, not $ 2^6 $!

Q: What if the bases were different numbers?

If bases are different (like $ 2^3 \times 3^4 $), you cannot combine them using exponent rules. The rule only works when the bases are exactly the same.

Q: How do I remember to add instead of multiply?

Think: "Same base, add the powers". You can also remember that $ 2^2 \times 2^2 = 4 \times 4 = 16 = 2^4 $, and 2 + 2 = 4, not 2 × 2!

Q: Can I solve this by calculating each power first?

You could calculate $ 2^3 = 8, 2^4 = 16 $, etc., then multiply 8 × 16 × 64 × 32. But using exponent rules is much faster and avoids huge number calculations!

Q: What's the final numerical value of 2¹⁸?

$ 2^{18} = 262,144 $. But for this problem, leaving the answer as $ 2^{18} $ in exponential form is the correct simplified answer.

Exponent Rules with Multiple Base Powers

Reduce the following equation:

$2^3\times2^4\times2^6\times2^5=$

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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 equal bases (A)

00:07 equals the same base raised to the sum of the exponents (N+M)

00:10 We will apply this formula to our exercise, each time for one multiplication

00:16 We'll maintain the base and add the exponents together

00:26 Calculate the exponents

00:32 Apply the formula once again

00:38 This is the solution

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Reduce the following equation:

$2^3\times2^4\times2^6\times2^5=$

Step-by-step solution

To reduce the expression $2^3 \times 2^4 \times 2^6 \times 2^5$ , we apply the rule of multiplication for exponents with the same base, which states that:

$a^m \times a^n = a^{m+n}$ .

Following this rule, we add up all the exponents together since they all have the same base, 2:

$3 + 4 + 6 + 5 = 18$ .

So, the expression reduces to $2^{18}$ .

Thus, the answer is $2^{18}$ .

Final Answer

$2^{18}$

Key Points to Remember

Essential concepts to master this topic

Base Rule: When multiplying powers with same base, add exponents
Technique: Add exponents: $3 + 4 + 6 + 5 = 18$
Check: Count all exponent terms match original expression ✓

Common Mistakes

Avoid these frequent errors

Multiplying exponents instead of adding them
Don't multiply 3 × 4 × 6 × 5 = 360! This gives $2^{360}$ which is completely wrong. Multiplication rule for same bases requires adding exponents, not multiplying them. Always add exponents: 3 + 4 + 6 + 5 = 18.

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 instead of multiplying them?

The rule $a^m \times a^n = a^{m+n}$ comes from repeated multiplication. For example, $2^3 \times 2^2 = (2 \times 2 \times 2) \times (2 \times 2) = 2^5$ , not $2^6$ !

What if the bases were different numbers?

If bases are different (like $2^3 \times 3^4$ ), you cannot combine them using exponent rules. The rule only works when the bases are exactly the same.

How do I remember to add instead of multiply?

Think: "Same base, add the powers". You can also remember that $2^2 \times 2^2 = 4 \times 4 = 16 = 2^4$ , and 2 + 2 = 4, not 2 × 2!

Can I solve this by calculating each power first?

You could calculate $2^3 = 8, 2^4 = 16$ , etc., then multiply 8 × 16 × 64 × 32. But using exponent rules is much faster and avoids huge number calculations!

What's the final numerical value of 2¹⁸?

$2^{18} = 262,144$ . But for this problem, leaving the answer as $2^{18}$ in exponential form is the correct simplified answer.

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