Reduce the Expression: 4^x × 4^2 × 4^a Using Exponent Laws

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

The multiplication rule for exponents says when you multiply powers with the same base, you add the exponents. Think of it this way: $ 4^2 \times 4^3 = (4 \times 4) \times (4 \times 4 \times 4) = 4^5 $, which is 2 + 3 = 5!

Q: What if the bases were different numbers?

If the bases are different, like $ 3^x \times 4^2 \times 5^a $, you cannot combine them using exponent rules. The rule only works when all bases are exactly the same.

Q: Can I use this rule with variables in the exponents?

Absolutely! Variables like x and a follow the same rules as numbers. Just treat them as you would any other exponent and add them together: x + 2 + a.

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

Multiplying same bases: ADD exponents ($ a^m \times a^n = a^{m+n} $)Power of a power: MULTIPLY exponents ($ (a^m)^n = a^{m \times n} $)

Q: What if one of the terms doesn't have an exponent showing?

Remember that $ 4 = 4^1 $! Any number without a visible exponent has an invisible exponent of 1. So $ 4 \times 4^x = 4^1 \times 4^x = 4^{1+x} $.

Exponent Laws with Multiple Base Terms

Reduce the following equation:

$4^x\times4^2\times4^a=$

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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 equal bases (A)

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

00:11 We will apply this formula to our exercise

00:15 We'll maintain the base and add the exponents together

00:24 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:

$4^x\times4^2\times4^a=$

Step-by-step solution

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

Step 1: Confirm the given expression is $4^x \times 4^2 \times 4^a$ .
Step 2: Apply the exponent rule for multiplication of powers: if $b^m \times b^n = b^{m+n}$ , use this with base 4.
Step 3: Add the exponents of each term.

Let's work through these steps:

Step 1: The expression we have is $4^x \times 4^2 \times 4^a$ .

Step 2: Since all parts of the product have the same base $4$ , we can use the rule for multiplying powers: $4^x \times 4^2 \times 4^a = 4^{x+2+a}$ .

Step 3: The simplified expression is obtained by adding the exponents: $x + 2 + a$ .

Therefore, the expression $4^x \times 4^2 \times 4^a$ simplifies to $4^{x+2+a}$ .

Final Answer

$4^{x+2+a}$

Key Points to Remember

Essential concepts to master this topic

Rule: When multiplying same bases, add their exponents together
Technique: $4^x \times 4^2 \times 4^a = 4^{x+2+a}$ by adding exponents
Check: Verify by substituting values: if x=1, a=3, then $4^1 \times 4^2 \times 4^3 = 4^6$ ✓

Common Mistakes

Avoid these frequent errors

Multiplying exponents instead of adding them
Don't multiply the exponents like $4^{x \times 2 \times a}$ = wrong answer! This confuses the multiplication rule with the power rule. Always add exponents when multiplying same bases: $4^x \times 4^2 \times 4^a = 4^{x+2+a}$ .

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 do I add the exponents instead of multiplying them?

The multiplication rule for exponents says when you multiply powers with the same base, you add the exponents. Think of it this way: $4^2 \times 4^3 = (4 \times 4) \times (4 \times 4 \times 4) = 4^5$ , which is 2 + 3 = 5!

What if the bases were different numbers?

If the bases are different, like $3^x \times 4^2 \times 5^a$ , you cannot combine them using exponent rules. The rule only works when all bases are exactly the same.

Can I use this rule with variables in the exponents?

Absolutely! Variables like x and a follow the same rules as numbers. Just treat them as you would any other exponent and add them together: x + 2 + a.

How do I remember when to add vs multiply exponents?

Multiplying same bases: ADD exponents ( $a^m \times a^n = a^{m+n}$ )
Power of a power: MULTIPLY exponents ( $(a^m)^n = a^{m \times n}$ )

What if one of the terms doesn't have an exponent showing?

Remember that $4 = 4^1$ ! Any number without a visible exponent has an invisible exponent of 1. So $4 \times 4^x = 4^1 \times 4^x = 4^{1+x}$ .

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