Expand the Exponential Expression: 4^(4+6) Step-by-Step

Q: Why can't I just add 4^4 + 4^6?

Because exponents represent repeated multiplication, not addition! $ 4^{4+6} $ means multiply 4 by itself 10 times, which equals $ 4^4 \times 4^6 $, not $ 4^4 + 4^6 $.

Q: Can I simplify 4+6 first to get 4^10?

Yes! $ 4^{4+6} = 4^{10} $ is correct, but the question asks you to expand the expression using exponent rules, so $ 4^4 \times 4^6 $ is the expected answer format.

Q: Does this rule work with different bases?

No! The rule $ a^{m+n} = a^m \times a^n $ only works when the bases are the same. For example, $ 3^2 \times 4^2 $ cannot be simplified using this rule.

Q: What if the exponent has subtraction instead?

Use the rule $ a^{m-n} = \frac{a^m}{a^n} $! For example, $ 4^{6-4} = \frac{4^6}{4^4} $. Subtraction in exponents means division of powers.

Q: How do I remember which operation to use?

Think of it this way: addition in the exponent means multiply the expanded terms, while subtraction in the exponent means divide the expanded terms. The base stays the same!

Exponent Rules with Sum in Power

Expand the following equation:

$4^{4+6}=$

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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)

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

00:12 We will apply this formula to our exercise, in reverse

00:18 We'll break it down into the product of the appropriate exponents

00:22 This is the solution

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Expand the following equation:

$4^{4+6}=$

Step-by-step solution

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

Step 1: Identify the base and the exponents in the expression
Step 2: Apply the exponent rule $a^{m+n} = a^m \times a^n$
Step 3: Rewrite the expression using the rule

Now, let's work through each step:
Step 1: The problem gives us the expression $(4)^{4+6}$ . Here, the base is 4, and the exponent is the sum $4 + 6$ .
Step 2: We'll apply the rule $a^{m+n} = a^m \times a^n$ , which allows us to write the expression as the product of two powers.
Step 3: According to the rule, $(4)^{4+6}$ becomes $(4)^4 \times (4)^6$ .

This means that $(4)^{4+6}$ expands to $(4)^4 \times (4)^6$ .

Therefore, the solution to the problem is $\boxed{4^4 \times 4^6}$ , corresponding to choice 4.

Final Answer

$4^4\times4^6$

Key Points to Remember

Essential concepts to master this topic

Rule: $a^{m+n} = a^m \times a^n$ splits sum into product
Technique: $4^{4+6} = 4^4 \times 4^6$ using the exponent rule
Check: Verify by calculating: $4^{10} = 4^4 \times 4^6 = 256 \times 4096$ ✓

Common Mistakes

Avoid these frequent errors

Adding exponents instead of multiplying bases
Don't think $4^{4+6} = 4^4 + 4^6$ = addition! This gives 256 + 4096 = 4352 instead of the correct 1,048,576. Always remember that $a^{m+n}$ means multiply the same base raised to separate powers.

Practice Quiz

Test your knowledge with interactive questions

$$

Simplify the following equation:

$5^8\times5^3=$

FAQ

Everything you need to know about this question

Why can't I just add 4^4 + 4^6?

Because exponents represent repeated multiplication, not addition! $4^{4+6}$ means multiply 4 by itself 10 times, which equals $4^4 \times 4^6$ , not $4^4 + 4^6$ .

Can I simplify 4+6 first to get 4^10?

Yes! $4^{4+6} = 4^{10}$ is correct, but the question asks you to expand the expression using exponent rules, so $4^4 \times 4^6$ is the expected answer format.

Does this rule work with different bases?

No! The rule $a^{m+n} = a^m \times a^n$ only works when the bases are the same. For example, $3^2 \times 4^2$ cannot be simplified using this rule.

What if the exponent has subtraction instead?

Use the rule $a^{m-n} = \frac{a^m}{a^n}$ ! For example, $4^{6-4} = \frac{4^6}{4^4}$ . Subtraction in exponents means division of powers.

How do I remember which operation to use?

Think of it this way: addition in the exponent means multiply the expanded terms, while subtraction in the exponent means divide the expanded terms. The base stays the same!

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