Simplify the Expression: 4^5 × 4 × 4^2 Using Exponent Rules

Q: Why does 4 equal 4^1? I don't see the exponent!

Any number without a visible exponent has an invisible exponent of 1. Just like how 4 = 4/1, we also have 4 = $ 4^1 $. This is a fundamental rule!

Q: Can I multiply the exponents instead of adding them?

No! Multiplying exponents is for powers of powers like $ (4^5)^2 $. When multiplying terms with the same base, you add the exponents.

Q: What if the bases were different numbers?

If bases are different (like $ 4^5 \times 3^2 $), you cannot combine them using exponent rules. The rule only works when the bases are identical!

Q: Do I need to calculate 4^8 for my final answer?

It depends on what the question asks! If it says simplify, then $ 4^8 $ is perfect. If it says evaluate, then calculate $ 4^8 = 65,536 $.

Q: How can I remember when to add vs multiply exponents?

Add exponents: Same base multiplication like $ a^m \times a^n = a^{m+n} $Multiply exponents: Power of a power like $ (a^m)^n = a^{m \times n} $

Exponent Rules with Multiple Terms

Simplify the following equation:

$4^5\times4\times4^2=$

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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 Any number raised to the power of 1 is always equal to itself

00:06 We'll apply this formula to our exercise, and raise it to the power of 1

00:12 According to the laws of exponents, the multiplication of powers with an equal base (A)

00:17 equals the same base raised to the sum of the exponents (N+M)

00:24 We'll apply this formula to our exercise

00:28 We'll maintain the base and proceed to add up the exponents

00:34 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:

$4^5\times4\times4^2=$

Step-by-step solution

To solve this simplification problem, we will apply the rules of exponents. Our steps are as follows:

Step 1: Identify the exponents of the base. We have $4^5$ , $4^1$ (since $4$ is equivalent to $4^1$ ), and $4^2$ .
Step 2: Use the property of exponents: $a^m \times a^n = a^{m+n}$ to combine powers of the same base.
Step 3: Calculate the sum of the exponents: $5 + 1 + 2 = 8$ .
Step 4: Express the simplified result in the form of a single power of 4: $4^{8}$ .

Therefore, the expression $4^5 \times 4 \times 4^2$ simplifies to $4^{5+1+2}$ , which further simplifies to $4^8$ .

Checking the multiple-choice options, the correct choice is: $4^{5+1+2}$ , aligning with our solution.

Final Answer

$4^{5+1+2}$

Key Points to Remember

Essential concepts to master this topic

Rule: When multiplying powers with same base, add the exponents
Technique: Recognize that 4 equals $4^1$ before adding: 5+1+2
Check: Verify $4^8 = 65,536$ matches original calculation ✓

Common Mistakes

Avoid these frequent errors

Forgetting that plain numbers have exponent 1
Don't treat 4 as having no exponent = missing terms in your addition! This gives $4^{5+2} = 4^7$ instead of the correct $4^8$ . Always remember that any number without an exponent has an invisible exponent of 1.

Practice Quiz

Test your knowledge with interactive questions

$112^0=\text{?}$

FAQ

Everything you need to know about this question

Why does 4 equal 4^1? I don't see the exponent!

Any number without a visible exponent has an invisible exponent of 1. Just like how 4 = 4/1, we also have 4 = $4^1$ . This is a fundamental rule!

Can I multiply the exponents instead of adding them?

No! Multiplying exponents is for powers of powers like $(4^5)^2$ . When multiplying terms with the same base, you add the exponents.

What if the bases were different numbers?

If bases are different (like $4^5 \times 3^2$ ), you cannot combine them using exponent rules. The rule only works when the bases are identical!

Do I need to calculate 4^8 for my final answer?

It depends on what the question asks! If it says simplify, then $4^8$ is perfect. If it says evaluate, then calculate $4^8 = 65,536$ .

How can I remember when to add vs multiply exponents?

Add exponents: Same base multiplication like $a^m \times a^n = a^{m+n}$
Multiply exponents: Power of a power like $(a^m)^n = a^{m \times n}$

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