Simplify 4^5 × 4^5: Multiplying Powers with Same Base

Q: Why do I add the exponents instead of multiplying them?

Think about what exponents mean: $ 4^5 = 4 \times 4 \times 4 \times 4 \times 4 $. When you multiply $ 4^5 \times 4^5 $, you're combining 10 total factors of 4, which equals $ 4^{10} $.

Q: What if the bases are different numbers?

The rule only works when the bases are exactly the same. For example, $ 3^2 \times 5^2 $ cannot be simplified using this rule because 3 and 5 are different bases.

Q: Does this work with negative exponents too?

Yes! The rule $ a^m \times a^n = a^{m+n} $ works with any exponents - positive, negative, or zero. Just add them like regular numbers.

Q: How can I remember this rule?

Think: "Same base, multiply powers? ADD the powers!" You can also remember that multiplication means combining groups of the same factor.

Q: What's the difference between this and dividing powers?

When multiplying same bases, you add exponents. When dividing same bases, you subtract exponents: $ \frac{4^7}{4^3} = 4^{7-3} = 4^4 $.

Exponent Rules with Same Base Multiplication

Simplify the following equation:

$4^5\times4^5=$

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Understand the problem

Simplify the following equation:

$4^5\times4^5=$

Step-by-step solution

To simplify the expression $4^5 \times 4^5$ , we will use the rule of exponents that states when multiplying two powers with the same base, you can add the exponents. This rule can be expressed as:

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

In this equation, both terms $4^5$ have the same base $4$ .

According to the multiplication of powers rule:

$4^5 \times 4^5 = 4^{5+5}$

Now, simply add the exponents:

$4^{5+5} = 4^{10}$

The simplified form of $4^5 \times 4^5$ is therefore $4^{10}$ .

Final Answer

$4^{10}$

Key Points to Remember

Essential concepts to master this topic

Rule: When multiplying powers with same base, add the exponents
Technique: $4^5 \times 4^5 = 4^{5+5} = 4^{10}$
Check: Verify base stays same and exponents were added correctly ✓

Common Mistakes

Avoid these frequent errors

Multiplying the exponents instead of adding them
Don't multiply exponents like $4^5 \times 4^5 = 4^{25}$ ! This gives a much larger, incorrect answer. Always add the exponents when multiplying powers with the same base.

Practice Quiz

Test your knowledge with interactive questions

$112^0=\text{?}$

FAQ

Everything you need to know about this question

Why do I add the exponents instead of multiplying them?

Think about what exponents mean: $4^5 = 4 \times 4 \times 4 \times 4 \times 4$ . When you multiply $4^5 \times 4^5$ , you're combining 10 total factors of 4, which equals $4^{10}$ .

What if the bases are different numbers?

The rule only works when the bases are exactly the same. For example, $3^2 \times 5^2$ cannot be simplified using this rule because 3 and 5 are different bases.

Does this work with negative exponents too?

Yes! The rule $a^m \times a^n = a^{m+n}$ works with any exponents - positive, negative, or zero. Just add them like regular numbers.

How can I remember this rule?

Think: "Same base, multiply powers? ADD the powers!" You can also remember that multiplication means combining groups of the same factor.

What's the difference between this and dividing powers?

When multiplying same bases, you add exponents. When dividing same bases, you subtract exponents: $\frac{4^7}{4^3} = 4^{7-3} = 4^4$ .

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